J I Nonnegative Tensors, Eigenvalues of Tensors Finsler ...statics.scnu.edu.cn/pics/maths/2013/1001/20131001095832740.pdf · Home Page Title Page JJ II J I Page 1 of 59 Go Back Full

Positive Definiteness . . .

Eigenvalues of . . .

Magnetic Resonance . . .

Solid Mechanics, . . .

Nonnegative Tensors, . . .

Finsler Geometry and . . .

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Eigenvalues of Tensorsand Their Applications

by

LIQUN QI

Department of Applied MathematicsThe Hong Kong Polytechnic University

http://math.suda.edu.cn







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Outline

� Positive Definiteness of Multivariate Forms

� Eigenvalues of Tensors and Numerical Multi-Linear Algebra

� Magnetic Resonance Imaging and Space Tensor Conic Programming

� Solid Mechanics, Quantum Physics and Biquadratic Optimization

� Nonnegative Tensors, Higher Order Markov Chains

and Spectral Hypergraph Theory

� Finsler Geometry and Relativity Theory








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1. Positive Definiteness of Multivariate FormsIn 2005, I defined eigenvalues and eigenvectors of a real symmetric tensor, andexplored their practical application in determining positive definiteness of aneven degree multivariate form. This work extended the classical concept ofeigenvalues of square matrices, forms an important part of numerical multi-linear algebra, and has found applications or links with automatic control, sta-tistical data analysis, optimization, magnetic resonance imaging, solid mechan-ics, quantum physics, higher order Markov chains, spectral hypergraph theory,Finsler geometry, relativity theory, etc.








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1.1. Quadratic Forms

Consider a quadratic form

f(x) ≡ xTAx =n∑

i,j=1

aijxixj,

where x = (x1, · · · , xn)T ∈ <n, A = (aij) is a symmetric matrix. If f(x) > 0

for all x ∈ <n, x 6= 0, then f and A are called positive definite. A complexnumber λ is called an eigenvalue of the matrix A if there is a nonzero complexvector x such that

Ax = λx.

A complex number is an eigenvalue of A if it is a root of the characteristicpolynomial of A. An n × n real symmetric matrix has n real eigenvalues. It ispositive definite if and only if all of its eigenvalues are positive. These are somebasic knowledge of linear algebra.Can these be extended to higher orders? Is there a need from practice for suchan extension?








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1.2. Homogeneous Polynomial Form

An n-dimensional homogeneous polynomial form of degree m, f(x), wherex ∈ <n, is equivalent to the tensor product of a symmetric n-dimensional tensorA = (ai1,··· ,im) of order m, and the rank-one tensor xm:

f(x) ≡ Axm :=n∑

i1,··· ,im=1

ai1,··· ,imxi1 · · ·xim.

The tensor A is called symmetric as its entries ai1,··· ,im are invariant under anypermutation of their indices. The tensor A is called positive definite (semidefi-nite) if f(x) > 0 (f(x) ≥ 0) for all x ∈ <n, x 6= 0. When m is even, the positivedefiniteness of such a homogeneous polynomial form f(x) plays an importantrole in the stability study of nonlinear autonomous systems via Liapunov’s di-rect method in Automatic Control. For n ≥ 3 and m ≥ 4, this issue is a hardproblem in mathematics.








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1.3. Study on the Positive Definiteness

[P1]. B.D. Anderson, N.K. Bose and E.I. Jury, “Output feedback stabilizationand related problems-solutions via decision methods”, IEEE Trans. Automat.Contr. AC20 (1975) 55-66.[P2]. N.K. Bose and P.S. Kamt, “Algorithm for stability test of multidimensionalfilters”, IEEE Trans. Acoust., Speech, Signal Processing, ASSP-22 (1974) 307-314.[P3]. N.K. Bose and A.R. Modaress, “General procedure for multivariable poly-nomial positivity with control applications”, IEEE Trans. Automat. Contr.AC21 (1976) 596-601.[P4]. N.K. Bose and R.W. Newcomb, “Tellegon’s theorem and multivariaterealizability theory”, Int. J. Electron. 36 (1974) 417-425.[P5]. M. Fu, “Comments on ‘A procedure for the positive definiteness of formsof even-order’ ”, IEEE Trans. Autom. Contr. 43 (1998) 1430.








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[P6]. M.A. Hasan and A.A. Hasan, “A procedure for the positive definiteness offorms of even-order”, IEEE Trans. Autom. Contr. 41 (1996) 615-617.[P7]. J.C. Hsu and A.U. Meyer, Modern Control Principles and Applications,McGraw-Hill, New York, 1968.[P8]. E.I. Jury and M. Mansour, “Positivity and nonnegativity conditions of aquartic equation and related problems” IEEE Trans. Automat. Contr. AC26(1981) 444-451.[P9]. W.H. Ku, “Explicit criterion for the positive definiteness of a generalquartic form”, IEEE Trans. Autom. Contr. 10 (1965) 372-373.[P10]. F. Wang and L. Qi, “Comments on ‘Explicit criterion for the positivedefiniteness of a general quartic form’ ”, IEEE Trans. Autom. Contr. 50 (2005)416- 418.








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2. Eigenvalues of Tensors and NumericalMulti-Linear Algebra

[1]. L. Qi, “Eigenvalues of a real supersymmetric tensor”, Journal of SymbolicComputation 40 (2005) 1302-1324,

defined eigenvalues and eigenvectors of a real symmetric tensor, and exploredtheir practical applications in determining positive definiteness of an even de-gree multivariate form.By the tensor product,Axm−1 for a vector x ∈ <n denotes a vector in<n, whoseith component is

(Axm−1)

i≡

n∑i2,··· ,im=1

ai,i2,··· ,imxi2 · · ·xim.








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We call a number λ ∈ C an eigenvalue of A if it and a nonzero vector x ∈ Cn

are solutions of the following homogeneous polynomial equation:(Axm−1)

i= λxm−1

i , ∀ i = 1, · · · , n. (1)

and call the solution x an eigenvector of A associated with the eigenvalue λ.We call an eigenvalue of A an H-eigenvalue of A if it has a real eigenvector x.An eigenvalue which is not an H-eigenvalue is called an N-eigenvalue. A realeigenvector associated with an H-eigenvalue is called an H-eigenvector.

The resultant of (1) is a one-dimensional polynomial of λ. We call it the char-acteristic polynomial of A.








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2.1. Theorem on Eigenvalues

Theorem 2.1 (Qi 2005)We have the following conclusions on eigenvalues of an mth order n-dimensional symmetric tensor A:(a). A number λ ∈ C is an eigenvalue of A if and only if it is a root of thecharacteristic polynomial φ.(b). The number of eigenvalues ofA is d = n(m−1)n−1. Their product is equalto det(A), the resultant of Axm−1 = 0.(c). The sum of all the eigenvalues of A is

(m− 1)n−1tr(A),

where tr(A) denotes the sum of the diagonal elements of A.(d). If m is even, then A always has H-eigenvalues. A is positive definite(positive semidefinite) if and only if all of its H-eigenvalues are positive (non-negative).(e). The eigenvalues of A lie in the following n disks:

|λ−ai,i,··· ,i| ≤∑

{|ai,i2,··· ,im| : i2, · · · , im = 1, · · · , n, {i2, · · · , im} 6= {i, · · · , i}},

for i = 1, · · · , n.








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2.2. An Example

Let m = 4 and n = 2. Assume that a1111 = a2222 = 1, a1112 = a1121 =a1211 = a2111 = a 6= 0 and other ai1,i2,i3,i4 = 0. Then from (1), we maydirectly find that A has four H-eigenvalues and two N-eigenvalues: a double H-eigenvalue λ1 = λ2 = 1 with an H-eigenvector x(1) = (0, 1)T , an H-eigenvalueλ3 = 1 + (27)

14a with an H-eigenvector x(3) = ((3)

14 , 1)T , an H-eigenvalue

λ4 = 1 − (27)14a with an H-eigenvector x(4) = ((3)

14 ,−1)T , an N-eigenvalue

λ5 = 1 + (27)14a√−1 with an N-eigenvector x(5) = ((3)

14 ,√−1)T , and an N-

eigenvalue λ6 = 1− (27)14a√−1 with an N-eigenvector x(6) = ((3)

14 ,−

√−1)T .

We see that the total number of eigenvalues is

d = n(m− 1)n−1 = 6,

the product of all the eigenvalues is 1− 27a4, and the sum of all the eigenvaluesis 6. On the other hand, by the resultant theory, we have

det(A) = 1− 27a4.








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This is equal to the product of all the eigenvalues. Also

(m− 1)n−1tr(A) = 6,

which is equal to the sum of all the eigenvalues. By the resultant theory, we alsohave

φ(λ) = (1− λ)2[(1− λ)4 − 27a4],

which has six roots λi for i = 1, · · · , 6 as indicated above.








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2.3. E-Eigenvalues

In the same paper, I defined another kind of eigenvalues for tensors. Their struc-ture is different from the structure described by Theorem 2.1. Their characteris-tic polynomial has a lower degree.

Suppose that A is an mth order n-dimensional symmetric tensor. We say acomplex number λ is an E-eigenvalue of A if there exists a complex vector xsuch that {

Axm−1 = λx,

xTx = 1.(2)

In this case, we say that x is an E-eigenvector of the tensor A associated withthe E-eigenvalue λ. If an E-eigenvalue has a real E-eigenvector, then we call ita Z-eigenvalue and call the real E-eigenvector a Z-eigenvector.








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2.4. The E-Characteristic Polynomial and Orthogonal Similarity

When m is even, the resultant of

Axm−1 − λ(xTx)m−2

2 x = 0

is a one dimensional polynomial of λ and is called the E-characteristic poly-nomial of A. We say that A is regular if the following system has no nonzerocomplex solutions: {

Axm−1 = 0,

xTx = 0.

Let P = (pij) be an n× n real matrix. Define B = PmA as another mth ordern-dimensional tensor with entries

bi1,i2,··· ,im =n∑

j1,j2,··· ,jm=1

pi1j1pi2j2

· · · pimjmaj1,j2,··· ,jm

.

If P is an orthogonal matrix, then we say thatA andB are orthogonally similar.








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2.5. Theorem on E-eigenvalues

Theorem 2.2 (Qi 2005)We have the following conclusions on E-eigenvalues of an mth order n-dimensional symmetric tensor A:(a). When A is regular, a complex number is an E-eigenvalue of A if and only ifit is a root of its E-characteristic polynomial.(b). Z-eigenvalues always exist. An even order symmetric tensor is positivedefinite if and only if all of its Z-eigenvalues are positive.(c). If A and B are orthogonally similar, then they have the same E-eigenvaluesand Z-eigenvalues.(d). If λ is the Z-eigenvalue of A with the largest absolute value and x is aZ-eigenvector associated with it, then λxm is the best rank-one approximationof A, i.e.,

‖A−λxm‖F =√‖A‖2

F − λ2 = min{‖A−αum‖F : α ∈ <, u ∈ <n, ‖u‖2 = 1},

where ‖ · ‖F is the Frobenius norm.








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2.6. Invariants of Tensors

Theorem 2.2 (c) implies that E-eigenvalues and Z-eigenvalues are invariant un-der co-ordinate system rotations. This is very important. Tensors are practicalphysical quantities in relativity theory, fluid dynamics, solid mechanics and elec-tromagnetism, etc. The concept of tensors was introduced by Gauss, Riemannand Christoffel, etc., in the 19th century in the study of differential geometry.In the very beginning of the 20th century, Ricci, Levi-Civita, etc., further de-veloped tensor analysis as a mathematical discipline. But it was Einstein whoapplied tensor analysis in his study of general relativity in 1916. This madetensor analysis an important tool in theoretical physics, continuum mechanicsand many other areas of science and engineering. The tensors in theoreticalphysics and continuum mechanics are physical quantities which are invariantunder co-ordinate system changes.In the study of tensors in physics and mechanics, an important concept is in-variant. A scalar associated with a tensor is an invariant of that tensor. Itdoes not change under co-ordinate system changes. Theorem 2.2 (c) impliesthat E-eigenvalues and Z-eigenvalues are invariants of the tensor. Later researchdemonstrate that these eigenvalues, in particular Z-eigenvalues, have practicaluses in physics and mechanics.

[2]. L. Qi, “Eigenvalues and invariants of tensors”, Journal of MathematicalAnalysis and Applications 325 (2007) 1363-1377,

also elaborated this.








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2.7. The Best Rank-One Approximation

Theorem 2.2 (d) indicates that Z-eigenvalues play an important role in the bestrank-one approximation. This also implies that Z-eigenvalues are significant inpractice. The best rank-one approximation of higher order tensors has extensiveengineering and statistical applications, such as Statistical Data Analysis. Thefollowing are some papers on this topic:

[B1]. L. De Lathauwer, B. De Moor and J. Vandwalle, “On the best rank-1 andrank-(R1, R2, · · · , Rn) approximation of higher order tensors”, SIAM J. MatrixAnal. Appl, 21 (2000) 1324-1342.

[B2]. E. Kofidies and Ph.A. Regalia, “On the best rank-1 approximation ofhigher order supersymmetric tensors”, SIAM J. Matrix Anal. Appl, 23 (2002)863-884.

[B3]. T. Zhang and G.H. Golub, “Rank-1 approximation to higher order ten-sors”, SIAM J. Matrix Anal. Appl, 23 (2001) 534-550.








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2.8. Lim’s work

Independently, at Stanford University in 2005, Gene Golub’s then Ph.D. studentLek-Heng Lim also defined eigenvalues for tensors in his paper:

[3]. L-H. Lim, “Singular values and eigenvalues of tensors: A variational ap-proach”, Proceedings of the 1st IEEE International Workshop on ComputationalAdvances in Multi-Sensor Adaptive Processing (CAMSAP), December 13-15,2005, pp. 129-132.

Lim (2005) defined eigenvalues for general real tensors in the real field. The l2

eigenvalues of tensors defined by Lim (2005) are Z-eigenvalues of Qi (2005),while the lk eigenvalues of tensors defined by Lim (2005) are H-eigenvaluesin Qi (2005). Notably, Lim (2005) proposed a multilinear generalization ofthe Perron-Frobenius theorem based upon the notion of lk eigenvalues (H-eigenvalues) of tensors.








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2.9. The Work of Chang, Pearson and Zhang

In 2009, Professor K.C. Chang at Peking University studied eigenvalues of realsymmetric tensors with his co-authors:

[4]. K.C. Chang, K. Pearson and T. Zhang, “On eigenvalues of real symmetrictensors”, J. Math. Anal. Appl. 350 (2009) 416-422.

In that paper, Professor Chang and his co-authors unified the definitions ofH-eigenvalues, Z-eigenvalues and D-eigenvalues, and show that for a realn-dimensional mth order symmetric tensor, there are at least n H-/Z-/D-eigenvalues.








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2.10. Calculation of Eigenvalues

The calculation of eigenvalues of a tensor is possible for small n. But it is NP-hard with respect to n when m ≥ 3. Some discussion on this can be found inthe following paper:

[5]. L. Qi, F. Wang and Y. Wang, “Z-eigenvalue methods for a global polynomialoptimization problem”, Mathematical Programming 118 (2009) 301-316.








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2.11. Numerical Multi-Linear Algebra

The work of eigenvalues of tensors received attention and support from GeneGolub, who was the leader of numerical linear algebra in the world. In 2005and 2007, he visited me twice, listening to me on the theory and applications ofeigenvalues of tensors. He invited me to Standford and Europe to participate inthe conferences organized by him and to give talks on eigenvalues of tensors.He pointed out that eigenvalues of tensors would be a very important part ofthe new subject — Numerical Multi-Linear Algebra. Unfortunately, Genepassed away on November 14, 2007. This was a great loss to the communityof numerical linear algebra and numerical multi-linear algebra. However, wewill do our best to build the new subject – Numerical Multi-Linear Algebra, inparticular, to explore more on eigenvalues of tensors and their applications.

A survey on numerical multi-linear algebra can be found in the following paper:

[6]. L. Qi, W. Sun and Y. Wang, “Numerical multilinear algebra and its applica-tions”, Frontiers of Mathematics in China 2 (2007) 501-526.








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2.12. Successive Tensor Decomposition

Another important part of numerical multi-linear algebra is tensor decompo-sition. The following paper discusses successive tensor decomposition basedupon Z-eigenvalues.

[7]. Y. Wang and L. Qi, “On the successive supersymmetric rank-1 decompo-sition of higher order supersymmetric tensors”, Numerical Linear Algebra withApplications 14 (2007) 503-519.

Let A be an mth order n-dimensional symmetric tensor. Let λ be its Z-eigenvalue with the largest absolute value, and x be a corresponding unit Z-eigenvector. According to Theorem 2.2 (d), λxm is the best rank-one approxi-mation of A. We also have

‖A‖2F = ‖A − λxm‖2

F + λ2.

Let A(0) = A. Suppose that we have A(k). Let λk be a Z-eigenvalue of A(k)

with the largest absolute value, and x(k) be a corresponding unit Z-eigenvector.Let

A(k+1) = A(k) − λk

(x(k)

)m

.

If A(K+1) = 0 for some K, then we have a finite successive decomposition ofA:

A =K∑

k=0

λk

(x(k)

)m

and ‖A‖2F =

K∑k=0

λ2k.








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We have the following theorems:

Theorem 2.3 (Wang and Qi 2007) If A(k+1) 6= 0 for all k, then we have aninfinite successive decomposition of A:

A =∞∑

k=0

λk

(x(k)

)m

and ‖A‖2F =

∞∑k=0

λ2k.

Theorem 2.4 (Wang and Qi 2007) In both the cases, we have

|λk+1| ≤ 2|λk|

for all k, and|λk+1| ≤ |λk|

if either m is even, and λk and λk+1 have the same sign, or x(k) and x(k+1) areorthogonal.

In general,|λk+1| ≤ |λk|

is not true.








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3. Magnetic Resonance Imaging and SpaceTensor Conic Programming

The Nobel Prize in Physiology or Medicine 2003: Paul C Lauterbur and PeterMansfield for their discoveries concerning “Magnetic Resonance Imaging”.

Summary: Imaging of human internal organs with exact and non-invasive meth-ods is very important for medical diagnosis, treatment and follow-up. Thisyear’s Nobel Laureates in Physiology or Medicine have made seminal discov-eries concerning the use of magnetic resonance to visualize different structures.These discoveries have led to the development of modern magnetic resonanceimaging, MRI, which represents a breakthrough in medical diagnostics and re-search.

Atomic nuclei in a strong magnetic field rotate with a frequency that is depen-dent on the strength of the magnetic field. Their energy can be increased ifthey absorb radio waves with the same frequency (resonance). When the atomicnuclei return to their previous energy level, radio waves are emitted. These dis-coveries were awarded the Nobel Prize in Physics in 1952. During the followingdecades, magnetic resonance was used mainly for studies of the chemical struc-ture of substances. In the beginning of the 1970s, this year’s Nobel Laureatesmade pioneering contributions, which later led to the applications of magneticresonance in medical imaging.








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3.1. Diffusion Tensor Imaging

Diffusion magnetic resonance imaging (D-MRI) has been developed in biomed-ical engineering for decades. It measures the apparent diffusivity of watermolecules in human or animal tissues, such as brain and blood, to acquire bi-ological and clinical information. In tissues, such as brain gray matter, wherethe measured apparent diffusivity is largely independent of the orientation of thetissue (i.e., isotropic), it is usually sufficient to characterize the diffusion char-acteristics with a single (scalar) apparent diffusion coefficient (ADC). However,in anisotropic media, such as skeletal and cardiac muscle and in white matter,where the measured diffusivity is known to depend upon the orientation of thetissue, no single ADC can characterize the orientation-dependent water mobilityin these tissues. Because of this, a diffusion tensor model was proposed yearsago to replace the diffusion scalar model. This resulted in Diffusion TensorImaging (DTI).








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3.2. Diffusion Tensor

A diffusion tensor D is a second order three dimensional fully symmetric tensor.Under a Cartesian laboratory co-ordinate system, it is represented by a real threedimensional symmetric matrix, which has six independent elements D = (dij)with dij = dji for i, j = 1, 2, 3. There is a relationship

ln[S(b)] = ln[S(0)]−3∑

i,j=1

bdijxixj. (3)

Here S(b) is the signal intensity at the echo time, x = (x1, x2, x3) is the unitdirection vector, satisfying

∑3i=1 x2

i = 1, the parameter b is given by

b = (γδg)2(∆− δ

3),

γ is the proton gyromagnetic ratio, ∆ is the separation time of the two diffu-sion gradients, δ is the duration of each gradient lobe. There are six unknownvariables dij in the formula (3). By applying the magnetic gradients in six ormore non-collinear, non-coplaner directions, one can solve (3) and get the sixindependent elements dij.








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3.3. Eigenvalues of Diffusion Tensor

However, such elements dij cannot be directly used for biological or clinicalanalysis, as they vary under different laboratory coordinate systems. Thus, af-ter obtaining the values of these six independent elements by MRI techniques,the biomedical engineering researchers will further calculate some characteristicquantities of this diffusion tensor D. These characteristic quantities are rotation-ally invariant, independent from the choice of the laboratory coordinate system.They include the three eigenvalues λ1 ≥ λ2 ≥ λ3 of D, the mean diffusivity(MD), the fractional anisotropy (FA), etc. The largest eigenvalue λ1 describesthe diffusion coefficient in the direction parallel to the fibres in the human tis-sue. The other two eigenvalues describe the diffusion coefficient in the directionperpendicular to the fibres in the human tissue. The mean diffusivity is

MD =λ1 + λ2 + λ3

3,

while the fractional anisotropy is

FA =

√3

2

√(λ1 −MD)2 + (λ2 −MD)2 + (λ3 −MD)2

λ21 + λ2

2 + λ23

,

where 0 ≤ FA ≤ 1. If FA = 0, the diffusion is isotropic. If FA = 1, thediffusion is anisotropic.








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3.4. Higher-Order Diffusion Tensor Imaging

However, DTI is known to have a limited capability in resolving multiple fibreorientations within one voxel. This is mainly because the probability densityfunction for random spin displacement is non-Gaussian in the confining en-vironment of biological tissues and, thus, the modeling of self-diffusion by asecond order tensor breaks down. Hence, researchers presented various HigherOrder Diffusion Tensor Imaging models to overcome this problem:

[H1]. J.H. Jensen, J.A. Helpern, A. Ramani, H. Lu and K. Kaczynski, “Diffu-sional kurtosis imaging: The quantification of non-Gaussian water diffusion bymeans of magnetic resonance imaging”, Magnetic Resonance in Medicine 53(2005) 1432-1440.[H2]. H. Lu, J.H. Jensen, A. Ramani and J.A. Helpern, “Three-dimensionalcharacterization of non-Gaussian water diffusion in humans using diffusion kur-tosis imaging”, NMR in Biomedicine 19 (2006) 236-247.[H3]. E. Ozarslan and T.H. Mareci, “Generalized diffusion tensor imaging andanalytical relationships between diffusion tensor imaging and high angular reso-lution diffusion imaging”, Magnetic Resonance in Medicine 50 (2003) 955-965.[H4]. D.S. Tuch, “Q-ball imaging”, Magnetic Resonance in Medicine 52 (2004)1358-1372,[H5]. D.S. Tuch, T.G. Reese, M.R. Wiegell, N.G. Makris, J.W. Belliveauand V.J. Wedeen, “High angular resolution diffusion imaging reveals intravoxelwhite matter fiber heterogeneity”, Magnetic Resonance in Medicine 48 (2002)454-459.








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3.5. Diffusion Kurtosis Tensor Imaging

The authors of [H1] and [H2] proposed to use a fourth order three dimensionalfully symmetric tensor W , called the diffusion kurtosis (DK) tensor, to describethe non-Gaussian behavior. Under a Cartesian laboratory co-ordinate system,it is represented by a real fourth order three dimensional fully symmetric array,which has fifteen independent elements W = (wijkl) with wijkl being invariantfor any permutation of its indices i, j, k, l = 1, 2, 3. The relationship (3) can befurther expanded (adding a second Taylor expansion term on b) to:

ln[S(b)] = ln[S(0)]−3∑

i,j=1

bdijxixj +1

6b2M2

D

3∑i,j,k,l=1

wijklxixjxkxl. (4)

There are fifteen unknown variables wijkl in the formula (4). By applying themagnetic gradients in fifteen or more non-collinear, non-coplaner directions,one can solve (4) and get the fifteen independent elements wijkl.

Again, the fifteen elements wijkl vary when the laboratory co-ordinate system isrotated. What are the coordinate system independent characteristic quantities ofthe DK tensor W ? Are there some type of eigenvalues of W , which can playa role here? Ed X. Wu and his group at Hong Kong University studied thesequestions. They searched by google possible papers on eigenvalues of higherorder tensors. They found my paper [1]. This resulted in a surprising e-mail tome in February, 2007, and consequently, some collaborative studies on diffusionkurtosis tensor imaging.








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3.6. D-Eigenvalues and Other Invariants

We proposed D-eigenvalues for the diffusion kurtosis tensor W and some otherinvariants to describe the DKI model.

[8]. L. Qi, Y. Wang and E.X. Wu, “D-eigenvalues of diffusion kurtosis tensors”,Journal of Computational and Applied Mathematics 221 (2008) 150-157.[9]. L. Qi, D. Han and E.X. Wu, “Principal invariants and inherent parameters ofdiffusion kurtosis tensors”, Journal of Mathematical Analysis and Applications349 (2009) 165-180.[10]. D. Han, L. Qi and E.X. Wu, “Extreme diffusion values for non-Gaussiandiffusions”, Optimization and Software 23 (2008) 703-716.[11]. E.S. Hui, M.M. Cheung, L. Qi and E.X. Wu, “Towards better MR charac-terization of neural tissues using directional diffusion kurtosis analysis”, Neu-roimage 42 (2008) 122-134.[12]. E.S. Hui, M.M. Cheung, L. Qi and E.X. Wu, “Advanced MR diffusioncharacterization of neural tissue using directional diffusion kurtosis analysis”,Conf. Proc. IEEE Eng. Med. Biol. Soc. 2008 (2008) 3941-3944.[13]. M.M. Cheung, E.S. Hui, K.C. Chan, J.A. Helpern, L. Qi, E.X. Wu, “Doesdiffusion kurtosis imaging lead to better neural tissue characterization? A rodentbrain maturation study”, Neuroimage 45 (2009) 386-392.[14]. X. Zhang, C. Ling, L. Qi and E.X. Wu “The measure of diffusion skewnessand kurtosis in magnetic resonance imaging”, to appear in: Pacific Journal ofOptimization.








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3.7. Paper by Bloy and Verma

The authors of [H3], [H4] and [H5] suggested to use a single higher even or-der tensor to replace the second order diffusion tensor in (3). In particular, in[H4], Tuch proposed to reconstruct the diffusion orientation distribution func-tion (ODF) of the underlying fiber population of a biological tissue. Experi-ments show that the QBI model may identify the underlying fiber directionswell in the multiple fiber situations.

In 2008, Bloy and Verma in their paper cited my paper [1], and proposed to useZ-eigenvalues to identify principal directions of fibres in the Q-balling model.

[15]. L. Bloy and R. Verma, “On computing the underlying fiber directions fromthe diffusion orientation distribution function”, in: Medical Image Computingand Computer-Assisted Intervention – MICCAI 2008, D. Metaxas, L. Axel, G.Fichtinger and G. Szekeley, eds., (Springer-Verlag, Berlin, 2008) pp. 1-8.








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3.8. Positive Semi-Definite Tensor Imaging

An intrinsic property of the diffusivity profile is positive semi-definite. Hence,the diffusion tensor, either second or higher order, must be positive semi-definite. For second order diffusion tensor, one may diagonalize the secondorder diffusion tensor and project it to the symmetric positive semi-definite coneby setting the negative eigenvalues to zero. Recently, some methods have beenproposed to preserve positive semi-definiteness for a fourth order diffusion ten-sor. None of them is comprehensive to work for arbitrary high order diffusiontensors.In

[16]. L. Qi, G. Yu and E.X. Wu, “Higher order positive semi-definite diffusiontensor imaging”, Department of Applied Mathematics, The Hong Kong Poly-technic University, March 2009,

we propose a comprehensive model to approximate the ADC profile by a posi-tive semi-definite diffusion tensor of either second or higher order. We call thismodel PSDT (positive semi-definite diffusion tensor).








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3.9. The PSDT Model

We use x = (x1, x2, x3)T to denote the magnetic field gradient direction. As-

sume that we use an mth order diffusion tensor. Then the diffusivity functioncan be expressed as

d(x) =m∑

i=0

m−i∑j=0

dijxi1x

j2x

m−i−j3 . (5)

A diffusivity function d can be regarded as an mth order symmetric tensor.Clearly, there are

N =m+1∑i=1

i =1

2(m + 1)(m + 2)

terms in (5). Hence, each diffusivity function can also be regarded as a vector din <N , indexed by ij, where j = 0, · · · , m− i, i = 0, · · · , m.

With the discussion above, it is ready now to formulate the PSDT (positive semi-definite tensor) model. It is as follows:

P (d∗) = min{P (d) : λmin(d) ≥ 0}, (6)

where P is a convex quadratic function of d, λmin is the smallest Z-eigenvalueof d.

Theorem 3.1 λmin(d) is a continuous concave function. Hence, PSDT (6) is aconvex optimization problem.








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3.10. Maps of Characteristic Quantities of PSDT

λmax

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 1 The map of λmax.








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MP SDT

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 2 The map of MPSDT .








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3.11. Space Tensor Conic Programming

Yinye Ye at Stanford University is a Visiting Chair Professor in our department.In April 2009, he visited our department. We discussed the PSDT model. Hesuggested to investigate this problem in a viewpoint of conic linear program-ming (CLP) problem. Actually, S = {d ∈ <N : λmin(d) ≥ 0} is a convex conein <N . We worked together on the following paper on Space Tensor ConicProgramming:

[17]. L. Qi and Y. Ye, “Space Tensor Conic Programming”, Department ofApplied Mathematics, The Hong Kong Polytechnic University, June 2009.

During the last two decades, major developments in convex optimization was fo-cusing on conic linear programming. Conic Linear programming (CLP) prob-lems include linear programming (LP) problems, semi-definite programming(SDP) problems and second-order cone programming (SOCP).

Space tensors appear in physics and mechanics. They are real physical entities.Mathematically, they are tensors of three dimension. All the mth order symmet-ric space tensors, where m is a positive even integer, form an N -dimensionalspace, where N = 1

2(m+1)(m+2). Using the method given in [5], we identifyin polynomial-time if an mth order symmetric space tensor is positive semi-definite or not.








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4. Solid Mechanics, Quantum Physics and Bi-quadratic Optimization

In Solid Mechanics, the elasticity tensor A = (aijkl) is partially symmetric inthe sense that for any i, j, k, l, we have aijkl = akjil = ailkj. We say that theyare strongly elliptic if and only if

f(x, y) ≡ Axyxy ≡n∑

i,j,k,l=1

aijklxiyjxkyl > 0,

for all unit vectors x, y ∈ <n, n = 2 or 3. For an isotropic material, someinequalities have been established to judge the strong ellipticity. See

[S1]. J.K. Knowles and E. Sternberg, “On the ellipticity of the equations ofnon-linear elastostatics for a special material”, J. Elasticity, 5 (1975) 341-361;[S2]. J.K. Knowles and E. Sternberg, “On the failure of ellipticity of the equa-tions for finite elastostatic plane strain”, Arch. Ration. Mech. Anal. 63 (1977)321-336;[S3]. H.C. Simpson and S.J. Spector, “On copositive matrices and strong ellip-ticity for isotrropic elstic materials”, Arch. Rational Mech. Anal. 84 (1983)55-68;[S4]. P. Rosakis, “Ellipticity and deformations with discontinuous deformationgradients in finite elastostatics”, Arch. Ration. Mech. Anal. 109 (1990) 1-37:[S5]. Y. Wang and M. Aron, “A reformulation of the strong ellipticity conditionsfor unconstrained hyperelastic media”, Journal of Elasticity 44 (1996) 89-96.








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4.1. M-Eigenvalues

In the following two papers, we studied conditions for strong ellipticity andintroduced M-eigenvalues for the ellipticity tensor A:

[18]. L. Qi, H.H. Dai and D. Han. “Conditions for strong ellipticity and M-eigenvalues”, Frontiers of Mathematics in China 4 (2009) 349-364;[19]. D. Han, H.H. Dai and L. Qi, “Conditions for strong ellipticity ofanisotropic elastic materials”, Journal of Elasticity 97 (2009) 1-13.

Denote A·yxy as a vector whose ith component is∑n

j,k,l=1 aijklyjxkyl, andAxyx· as a vector whose lth component is

∑ni,j,k=1 aijklxiyjxk. If λ ∈ <, x ∈

<n and y ∈ <n satisfy {A·yxy = λx, Axyx· = λy,

xTx = 1, yTy = 1,(7)

we call λ an M-eigenvalue of A, and call x and y left and right M-eigenvectorsof A, associated with the M-eigenvalue λ. Here, the letter “M” stands for me-chanics.

Theorem 4.1 M-eigenvalues always exist. The strong ellipticity condition holdsif and only if the smallest M-eigenvalue of the elasticity tensor A is positive.








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4.2. The Entanglement Problem in Quantum Physics

In the definition of M-eigenvalues, x, y ∈ <n and n = 3. If we allow x ∈<n, y ∈ <m, n and m can be any integers, then the smallest M-eigenvalueproblem has an application in the entanglement problem in Quantum Physics.

The entanglement problem is to determine whether a quantum state is separableor inseparable (entangled), or to check whether an mn×mn symmetric matrixA � 0 can be decomposed as a convex combination of tensor products of n andm dimensional vectors. It has fundamental importance in quantum science andhas attracted much attention since the pioneer work of Einstein, Podolsky andRosen (1935) and Schrodinger (1935):

[Q1]. A. Einstein, B. Podolsky and N. Rosen, “Can quantum-mechanical de-scription of physical reality be considered complete?” Physical Review 47(1935) 777-780.

[Q2]. E. Schrodinger, “Die gegenwartige situation in der quantenmechanik,”Naturwissenschaften 23 (1935) 807-812, 823-828, 844-849.








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4.3. The Biquadratic Optimization

The smallest M-eigenvalue problem can be formulated as the following Bi-quadratic Optimization problem

minx∈<n,y∈<m

f(x, y) =n∑

i,k=1

m∑j,l=1

aijklxiyjxkyl

subject to x>x = 1, y> = 1,(8)

where the coefficients aijkl satisfy the symmetric property: aijkl = akjil = ailkj

for i, k = 1, · · · , n and j, l = 1, · · · , m. Denote A := (aijkl). Then A is afourth order partially symmetric tensor.

If one of n and m is fixed, then the biquadratic optimization problem is polyno-mial time solvable. Otherwise, it is NP-hard.








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We discussed the biquadratic optimization problem in the following four papers:

[20]. Y. Wang, L. Qi and X. Zhang, “A practical method for computing thelargest M-eigenvalue of a fourth-order partially symmetric tensor”, NumericalLinear Algebra with Applications 16 (2009) 589-601.

[21]. C. Ling, J. Nie, L. Qi and Y. Ye, “Bi-quadratic optimization over unitspheres and semidefinite programming relaxations”, SIAM Journal on Opti-mization 20 (2009) 1286-1310.

[22]. D. Han and L. Qi, “A successive approximation method for quantum sep-arability”, Department od Applied Mathematics, The Hong Kong PolytechnicUniversity, February 2009.

[23]. X. Zhang, C. Ling and L. Qi, “Semidefinite relaxation bounds for bi-quadratic optimization problems with quadratic constraints”, Department odApplied Mathematics, The Hong Kong Polytechnic University, July 2009.

In [21], Ling, Nie, Qi and Ye gave numerical examples for biquadratic optimiza-tion problems with n = 300 and m = 600.








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5. Nonnegative Tensors, Higher OrderMarkov Chains and Spectral HypergraphTheory

After Lim (2005), there are following three papers on eigenvalues of Nonnega-tive Tensors:

[24]. K. C. Chang, K. Pearson and T. Zhang, Perron-Frobenius theorem fornonnegative tensors, Commu. Math. Sci. 6 (2008) 507-520.

[25]. M. Ng, L. Qi and G. Zhou, “Finding the largest eigenvalue of a non-negative tensor”, SIAM Journal on Matrix Analysis and Applications 31 (2009)1090-1099.

[26]. Y. Yang and Q. Yang, “Further results for Perron-Frobenius Theoremfor nonnegative tensors”, School of Mathematical Sciences, Nankai University,2009.








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5.1. The Work of Chang, Pearson and Zhang

First of all, Chang, Pearson and Zhang (2008) extended the definition ofeigenvalues of symmetric tensors given by Qi (2005) to general mth order n-dimensional tensor:

Definition 5.1 Let A be an m-order n-dimensional tensor and C be the set ofall complex numbers. Assume that Axm−1 is not identical to zero. We say(λ, x) ∈ C × (Cn\{0}) is an eigenvalue-eigenvector of A if

Axm−1 = λx[m−1]. (9)

Here, x[α] = [xα1 , xα

2 , ..., xαn]T .

Then they defined irreducible tensors:

Definition 5.2 An m-order n-dimensional tensor A is called reducible if thereexists a nonempty proper index subset I ⊂ {1, 2, ..., n} such that

Ai1i2...im = 0, ∀i1 ∈ I, ∀i2, ..., im /∈ I.

If A is not reducible, then we call A irreducible.








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5.2. The Perron-Frobenius Theorem for Nonnegative Tensors

The Perron-Frobenius Theorem for non-negative tensors is related to measuringhigher order connectivity in linked objects and hypergraphs. In the following,we state the Perron-Frobenius Theorem for non-negative tensors.

Theorem 5.1 (Chang, Pearson and Zhang 2008) If A is an irreducible non-negative tensor of order m and dimension n, then there exist λ0 > 0 and x0 >0, x0 ∈ Rn such that

Axm−10 = λ0x

[m−1]0 . (10)

Moreover, if λ is an eigenvalue with a non-negative eigenvector, then λ = λ0. Ifλ is an eigenvalue of A, then |λ| ≤ λ0.








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5.3. The Collatz Minimax Theorem for Nonnegative Tensors

Chang, Pearson and Zhang (2008) extended the well-known Collatz minimaxtheorem for irreducible non-negative matrices to irreducible non-negative ten-sors.Let P = {x ∈ Rn : xi ≥ 0, 1 ≤ i ≤ n} and int(P ) = {x ∈ Rn : xi > 0, 1 ≤i ≤ n}. We have the following minimax theorem for irreducible non-negativetensors.

Theorem 5.2 (Chang, Pearson and Zhang 2008) Assume that A is an irre-ducible non-negative tensor of order m and dimension n. Then

Minx∈int(P )Maxxi>0(Axm−1)i

xm−1i

= λ0 = Maxx∈int(P )Minxi>0(Axm−1)i

xm−1i

, (11)

where λ0 is the unique positive eigenvalue corresponding to the positive eigen-vector.








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5.4. The Work of Ng, Qi and Zhou

Let A be an irreducible non-negative tensor of order m and dimension n andx(0) ∈ <n be an arbitrary positive vector. Let y(0) = A

(x(0)

)m−1. Define

x(1) =

(y(0)

)[ 1m−1]∥∥∥∥(

y(0))[ 1

m−1]∥∥∥∥, y(1) = A

(x(1)

)m−1,

x(2) =

(y(1)

)[ 1m−1]∥∥∥∥(

y(1))[ 1

m−1]∥∥∥∥, y(2) = A

(x(2)

)m−1,

...

x(k+1) =

(y(k)

)[ 1m−1]∥∥∥∥(

y(k))[ 1

m−1]∥∥∥∥, y(k+1) = A

(x(k+1)

)m−1, k ≥ 2,

...

and let

λk = minx

(k)i >0

(A(x(k))m−1

)i(

x(k)i

)m−1 , λk = maxx

(k)i >0

(A(x(k))m−1

)i(

x(k)i

)m−1 , k = 1, 2, ...








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Theorem 5.3 (Ng, Qi and Zhou (2009) Assume that λ0 is the unique positiveeigenvalue corresponding to a non-negative eigenvector. Then, in the abovesetting, we have

λ1 ≤ λ2 ≤ · · · ≤ λ0 ≤ · · · ≤ λ2 ≤ λ1.

We know that general eigenvalue problem for higher order tensors, even in thesymmetric case, is NP-hard. Theorem 5.3 informs us that in the nonnegativecase, the situation is much better. Further exploration on the convergence theoryof finding the largest eigenvalue of a nonnegative tensor is going on.

Ng, Qi and Zhou (2009) further applied their method to compute the probabilitydistribution of a Higher order Markov Chain.








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5.5. The Work of Yang and Yang

Theorem 5.4 (Yang and Yang 2009) Call ρ(A) the spectral radius of tensorA if it equals the largest absolute eigenvalue of A, i.e. ρ(A) = max{|λ| :λ is an eigenvalue of A}.IfA is a nonnegative tensor of order m and dimension n, then ρ(A) is an eigen-value of A with a nonnegative eigenvector y ∈ <n

+ corresponding to it.Suppose furthermore that A is a positive order m dimension n tensor. If λ is aneigenvalue of A except ρ(A), then ρ(A) > |λ|.








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5.6. Spectral Hypergraph Theory

Lim [3] pointed out that another potential application of eigenvalues of tensorsis on hypergraphs. A graph can be described by its adjacency matrix. Theproperties of eigenvalues of the adjacency matrix of a graph are related to theproperties of the graph. This is the topic of spectral graph theory. A hypergraphcan be described by a (0, 1)- symmetric tensor, which is called its adjacencytensor. Are the properties of the eigenvalues of the adjacency tensor of a hyper-graph related to the properties of the hypergraph? This can be investigated.

There was an article cited in the literature:

[27]. P. Drineas and L-H. Lim, “A multilinear spectral theory of hypergraphsand expander hypergraphs”, Preprint, 2005.

Recently, a new paper by some Italian researchers on this topic appeared:

[28]. S.R. Bulo and M. Pelillo, “New bounds on the clique number of graphsbased on spectral hypergraph theory”, Dipartimento di Informatica, Universitadi Venezia, 2009.

Note that the (0, 1)-adjacency tensor is a nonnegative tensor. Hence, the spectralhypergraph theory is connected with eigenvalues of nonnegative tensors.








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6. Finsler Geometry and Relativity TheoryMy second paper on eigenvalues of tensors is on their geometric properties.

[29]. L. Qi, “Rank and eigenvalues of a supersymmetric tensor, the multivariatehomogeneous polynomial and the algebraic hypersurface it defines”, Journal ofSymbolic Computation 41 (2006) 1309-1327.

This paper deals with the orthogonal classification problem of real hypersurfacesgiven by an equation of the form

S = {x ∈ <n : f(x) = Axm = c},

where A is a symmetric tensor, through the rank, the Z-eigenvalues and theasymptotic directions.








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6.1. Balan’s Work

Recently, a Romanian researcher, who’s fields are Differential Geometry andRelativity Theory, stepped in the discussion of eigenvalues of tensors:

[30]. V. Balan, “Spectral properties and applications of numerical multi-linearalgebra of m-root structures”, in: Hypercomplex Numbers in Geometry andPhysics, ed., “Mozet”, Russia, 2 (2008) 101-107.

[31]. V. Balan, “Numerical multilinear algebra of symmetric m-root structures:Spectral properties and applications”, to appear in: Symmetry, (2009).

[32]. V. Balan, “Spectra of symmetric tensors and m-root Finsler geometrymodels”, Talk at The Third International Conference on Structured Matricesand Tensors, Hong Kong, January 2010.








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6.2. Finsler Geometry, Relativity Theory and Eigenvalues of Tensors

The first Finsler metric was pointed out by B. Riemann in his famous lec-ture (habilitation address) in 1854. The theory of Finsler Geometry was laiddown by Paul Finsler in his Doctoral Dissertation at Gottingen in 1918, withCaratheodory as his scientific supervisor. As S.S. Chern commented in 1996,Finsler geometry is just Riemannian geometry without the quadratic restriction.

The significance of Finsler geometry is that it gave the geometric foundation ofmodern Relativity Theory.

Balan’s work show that there is a close relationship between Finsler metrics ofBerwald-Moor type, and multilinear spectral theory based upon eigenvalues ofhigher order tensors. Balan used the results of my papers [1] and [29] to analyzespectral properties of m-root models, geometric relevance of spectral equations,degeneracy sets, asymptotic rays, base indices and based rank one approxima-tion of various Berwald-Moor tensors, Chernov tensors and Bogoslovski ten-sors.








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6.3. Books on Finsler Geometry

[F1]. G.S. Asanov, Finsler Geometry, Relativity and Gauge Theories, KluwerAcademic publishers, Dordrecht, 1996.

[F2]. D. Bao, S.S. Chern and Z. Shen, An Introduction to Riemann-FinslerGeometry, Springer Verlag, Berlin, 2000.

[F3]. S.S. Chern and Z. Shen, Riemann-Finsler Geometry, World Scientific,Singapore, 2005.

[F4]. X. Mo, An Introduction to Finsler Geometry, World Scientific, Singapore,2006.

[F5]. Z. Shen, Lectures on Finsler Geometry, World Scientific, Singapore, 2001.








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6.4. Tangent Bundle

Let M be a smooth (C4 is enough) m-dimensional manifold. For a point x ∈M , denote by TxM the tangent space of M at x. The tangent bundle TM isthe union of tangent spaces with a natural differential structure,

TM :⋃

x∈M

TxM.

Denote the elements in TM by (x, y) where y ∈ TxM .

The slit tangent space of M at x is defined by

TxM0 := TxM \ {0}.

The slit tangent bundle TM0 is defined by

TM0 := TM \ {0}.








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6.5. Pseduo-Finsler Metric

A continuous function F (x, y) defined on TM is called a pseduo-Finsler met-ric of index q if the following three conditions are satisfied:(a). F is smooth (C∞) on TM0.(b). F is positive homogenous in y, i.e., we have

F (x, ky) = kF (x, y), ∀k > 0.

(c). The metric tensor

gij(x, y) =1

2

∂F 2

∂yi∂yj

is non-degenerate, with constant signature on TM0, i.e., it has q negative eigen-values and m− q positive eigenvalues for all (x, y) ∈ TM0.

Then we say that Fm = (M, F ) is a pseudo-Finsler manifold of index q.

If q = 0, then F is called a Finsler metric and Fm is called a Finsler manifold.








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6.6. Main Finsler Metrics

• Riemannian metric (the quadratic case):

FRiem(x, y) =√

aij(x)yiyj.

From now on, we use the Einstein convention. Hence

aij(x)yiyj =m∑

i,j=1

aij(x)yiyj.

• Randers metric:

FRanders(x, y) = FRiem(x, y) + bk(x)yk.

• Riemann-Finsler metric:

FRF (x, y) = 4

√y4

1 + y42 + y4

3 + y44.








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6.7. Finsler M-Root Symmetric Metric

• the Shimada metric (1979)

FShimada(x, y) = m√

ai1i2···imyi1 · · · yim.

Here A = (ai1i2···im) is an mth order m-dimensional symmetric tensor.

• the m-root metric of Berwald-Moor type:

FBerwald−Moor(x, y) = m√|y1 · · · ym|.

This includes the Berwald metrics, the Chernov metrics and the Bo-goslovski metric, etc. Recently, the Berwald-Moor metrics are consideredby physicists as an important subject for a possible model of space-time.








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6.8. The M-Root Finsler Geometry and Multi-Linear Spectral The-ory

Each m-root Finsler metric is associated by a high order symmetric tensor.Hence, the eigenvalues of these tensors are associated with properties of suchFinsler metrics. Balan (2008, 2009, 2010) started the research at this direction.








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6.9. Final Remarks

We are stepping to a new territory of applied mathematics. We are developingcomputational multilinear algebraic methods for higher-order tensors, whichare parallel to the matrix theory. We are studying mathematical properties oftensors, such as eigenvalues and characteristic polynomials of tensors, tensordecomposition and tensor approximation, etc. The new subject has found ap-plications in or links with data analysis, automatic control, magnetic resonanceimaging, optimization, solid mechanics, quantum physics, higher order Markovchains, spectral hypergraph theory, Finsler geometry and relativity theory, etc.Further serious exploration on these aspects is needed. I hope that more re-searchers will join us to explore this new field.

My website on this field:http://www.polyu.edu.hk/ama/staff/new/qilq/TensorComp.htm


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