Introduction One Simple Algorithm Applications

Inverse Eigenvalue ProblemsConstructing Matrices with Prescribed Eigenvalues

N. Jackson

Department of MathematicsCollege of the Redwoods

Math 45 Term Project, Fall 2010

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Outline

IntroductionEigenvalues and EigenvectorsInverse Eigenvalue Problems (IEP’s)

One Simple AlgorithmHeuvers’ AlgorithmProofAn ExampleBenefits and Drawbacks

Applications

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Eigenvalues and Eigenvectors

What are Eigenvalues and Eigenvectors?

I An eigenvalue is “any number such that a given squarematrix minus that number times the identity matrix has azero determinant” [2].

I Ax = λx

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Eigenvalues and Eigenvectors

What are Eigenvalues and Eigenvectors?

I An eigenvalue is “any number such that a given squarematrix minus that number times the identity matrix has azero determinant” [2].

I Ax = λx

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Inverse Eigenvalue Problems (IEP’s)

Inverse Eigenvalue Problems (IEP’s)

I A well-studied yet continually developing branch of LinearAlgebra concerning construction of matrices from spectraldata.[3]

I Two basic components: solvability and computability.

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Inverse Eigenvalue Problems (IEP’s)

Inverse Eigenvalue Problems (IEP’s)

I A well-studied yet continually developing branch of LinearAlgebra concerning construction of matrices from spectraldata.[3]

I Two basic components: solvability and computability.

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Heuvers’ Algorithm

Konrad Heuvers’ AlgorithmSymmetric Matrices with Prescribed Eigenvalues and Eigenvectors

I Let {p1,p2, . . . ,pn} be an arbitrary orthonormal basis forRn. These will become the eigenvectors.

I Let λ1, λ2, ..., λn be n arbitrary real numbers (the desiredeigenvalues) and τ be any real number such that τ ≤ λj forj = 1,2, ...,n.

I Define µj =√λj − τ and bj = µjpj , and let B be the matrix

comprised of the column vectors b1,b2, ...bn.I Let S be the matrix S = BBT + τ I, a symmetric matrix with

the above eigenvectors and eigenvalues.

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Heuvers’ Algorithm

Konrad Heuvers’ AlgorithmSymmetric Matrices with Prescribed Eigenvalues and Eigenvectors

I Let {p1,p2, . . . ,pn} be an arbitrary orthonormal basis forRn. These will become the eigenvectors.

I Let λ1, λ2, ..., λn be n arbitrary real numbers (the desiredeigenvalues) and τ be any real number such that τ ≤ λj forj = 1,2, ...,n.

I Define µj =√λj − τ and bj = µjpj , and let B be the matrix

comprised of the column vectors b1,b2, ...bn.I Let S be the matrix S = BBT + τ I, a symmetric matrix with

the above eigenvectors and eigenvalues.

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Heuvers’ Algorithm

Konrad Heuvers’ AlgorithmSymmetric Matrices with Prescribed Eigenvalues and Eigenvectors

I Let {p1,p2, . . . ,pn} be an arbitrary orthonormal basis forRn. These will become the eigenvectors.

I Let λ1, λ2, ..., λn be n arbitrary real numbers (the desiredeigenvalues) and τ be any real number such that τ ≤ λj forj = 1,2, ...,n.

I Define µj =√λj − τ and bj = µjpj , and let B be the matrix

comprised of the column vectors b1,b2, ...bn.I Let S be the matrix S = BBT + τ I, a symmetric matrix with

the above eigenvectors and eigenvalues.

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Heuvers’ Algorithm

Konrad Heuvers’ AlgorithmSymmetric Matrices with Prescribed Eigenvalues and Eigenvectors

I Let {p1,p2, . . . ,pn} be an arbitrary orthonormal basis forRn. These will become the eigenvectors.

I Let λ1, λ2, ..., λn be n arbitrary real numbers (the desiredeigenvalues) and τ be any real number such that τ ≤ λj forj = 1,2, ...,n.

I Define µj =√λj − τ and bj = µjpj , and let B be the matrix

comprised of the column vectors b1,b2, ...bn.I Let S be the matrix S = BBT + τ I, a symmetric matrix with

the above eigenvectors and eigenvalues.

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Proof

Proof of Heuvers’ Algorithm

I The columns of B are(b1,b2, . . . ,bn) = (µ1p1, µ2p2, . . . , µnpn).

I The rows of BT are of the form µipTi .

I It must be shown that Spj = λjpj .

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Proof

Proof of Heuvers’ Algorithm

I The columns of B are(b1,b2, . . . ,bn) = (µ1p1, µ2p2, . . . , µnpn).

I The rows of BT are of the form µipTi .

I It must be shown that Spj = λjpj .

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Proof

Proof of Heuvers’ Algorithm

I The columns of B are(b1,b2, . . . ,bn) = (µ1p1, µ2p2, . . . , µnpn).

I The rows of BT are of the form µipTi .

I It must be shown that Spj = λjpj .

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Proof

Proof (cont., 2)

Spj = (BBT + τ I)pj

= BBT pj + τpj

= [µ1p1, µ2p2, . . . , µnpn]

µ1pT

1µ2pT

2...

µnpTn

pj + τpj

= [µ1p1, µ2p2, . . . , µnpn]

µ1pT

1 pjµ2pT

2 pj...

µnpTn pj

+ τpj

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Proof

Proof (cont., 3)

I Column vector all zeros except pj dotted with itself is one.I

. . . = [µ1p1, µ2p2, . . . , µnpn]

µ1pT

1 pjµ2pT

2 pj...

µnpTn pj

+ τpj

= [µ1p1, µ2p2, . . . , µnpn]

0...µj...0

+ τpj

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Proof

Proof (cont., 3)

I Column vector all zeros except pj dotted with itself is one.I

. . . = [µ1p1, µ2p2, . . . , µnpn]

µ1pT

1 pjµ2pT

2 pj...

µnpTn pj

+ τpj

= [µ1p1, µ2p2, . . . , µnpn]

0...µj...0

+ τpj

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Proof

Proof (cont., 4)

I

= µ2j pj + τpj

= (µ2j + τ)pj

= λjpj

I We have shown that Spj = λjpj , therefore each vector pjand corresponding scalar λj are an eigenvector andeigenvalue for the matrix S.

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Proof

Proof (cont., 4)

I

= µ2j pj + τpj

= (µ2j + τ)pj

= λjpj

I We have shown that Spj = λjpj , therefore each vector pjand corresponding scalar λj are an eigenvector andeigenvalue for the matrix S.

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

An Example

An Example

I Arbitrary orthonormal basis for R2:

BR2 =

{[√2/2√

2/2

],

[−√

2/2√

2/2

]}

I For simplicity of computing µ1 and µ2, we’ll choose λ1 = 2,λ2 = 5, and τ = 1.

I

µ1 =√λ1 − τ =

√2− 1 = 1

µ2 =√λ2 − τ =

√5− 1 = 2

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

An Example

An Example

I Arbitrary orthonormal basis for R2:

BR2 =

{[√2/2√

2/2

],

[−√

2/2√

2/2

]}

I For simplicity of computing µ1 and µ2, we’ll choose λ1 = 2,λ2 = 5, and τ = 1.

I

µ1 =√λ1 − τ =

√2− 1 = 1

µ2 =√λ2 − τ =

√5− 1 = 2

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

An Example

An Example

I Arbitrary orthonormal basis for R2:

BR2 =

{[√2/2√

2/2

],

[−√

2/2√

2/2

]}

I For simplicity of computing µ1 and µ2, we’ll choose λ1 = 2,λ2 = 5, and τ = 1.

I

µ1 =√λ1 − τ =

√2− 1 = 1

µ2 =√λ2 − τ =

√5− 1 = 2

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

An Example

An Example (cont., 2)

I Create the matrix B, composed of the columns b1 and b2:I

B = [b1,b2]

= [µ1p1, µ2p2]

=

[1

[√2/2√

2/2

],2

[−√

2/2√

2/2

]]

=

[√2/2 −

√2

√2/2

√2

]

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

An Example

An Example (cont., 2)

I Create the matrix B, composed of the columns b1 and b2:I

B = [b1,b2]

= [µ1p1, µ2p2]

=

[1

[√2/2√

2/2

],2

[−√

2/2√

2/2

]]

=

[√2/2 −

√2

√2/2

√2

]

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

An Example

An Example (cont., 3)

Now we can create our matrix S:

S = BBT + τ I

=

[√2/2 −

√2

√2/2

√2

][√2/2

√2/2

−√

2√

2

]+ 1

[1 00 1

]

=

[5/2 −3/2

−3/2 5/2

]+

[1 00 1

]

=

[7/2 −3/2

3/2 7/2

]

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

An Example

An Example (cont., 4)

Check this solution by first finding the eigenvalues of S:

det(S − λI) = 0∣∣∣∣∣7/2− λ −3/2

−3/2 7/2− λ

∣∣∣∣∣ = 0

(7/2− λ)2 − (−3/2)2 = 0

49/4− 7λ+ λ2 − 9/4 = 0

λ2 − 7λ+ 10 = 0(λ− 2)(λ− 5) = 0

λ = 2,5

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

An Example

An Example (cont., 5)

...and then by finding the corresponding eigenvectors:I

(S − 2I)x = 0[3/2 −3/2−3/2 3/2

] [xy

]=

[00

][xy

]=

[11

]I

(S − 5I)x = 0[−3/2 −3/2−3/2 −3/2

] [xy

]=

[00

][xy

]=

[−11

]Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

An Example

An Example (cont., 5)

...and then by finding the corresponding eigenvectors:I

(S − 2I)x = 0[3/2 −3/2−3/2 3/2

] [xy

]=

[00

][xy

]=

[11

]I

(S − 5I)x = 0[−3/2 −3/2−3/2 −3/2

] [xy

]=

[00

][xy

]=

[−11

]Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

An Example

An Example (cont., 6)

I Note that the eigenvectors of the S we created with thealgorithm are scalar multiples of the eigenvectors wechose beforehand.

I Since the nullspaces of S − 2I and S − 5I are both closedunder scalar multiplication, the eigenvectors we foundconfirm the validity of the algorithm.

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

An Example

An Example (cont., 6)

I Note that the eigenvectors of the S we created with thealgorithm are scalar multiples of the eigenvectors wechose beforehand.

I Since the nullspaces of S − 2I and S − 5I are both closedunder scalar multiplication, the eigenvectors we foundconfirm the validity of the algorithm.

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

An Example

An Example (cont., 7)

x

y [11

][−11

][√

2/2√2/2

][−√

2/2√2/2

]

Figure: Eigenvectors of matrix S.

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Benefits and Drawbacks

Benefits and Drawbacksof Heuvers’ Algorithm

I Simple to understand and compute.I Always creates symmetric matrices, must normalize

eigenvectors first.

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Benefits and Drawbacks

Benefits and Drawbacksof Heuvers’ Algorithm

I Simple to understand and compute.I Always creates symmetric matrices, must normalize

eigenvectors first.

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Applicationsof Inverse Eigenvalue Problems

I Found in applications where goal is finding physicalparameters of a system based on known behavior orconstructing a system with physical parameters resulting ina desired dynamical behavior [3].

I Particle physicsI Molecular spectroscopyI Geophysics

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Applicationsof Inverse Eigenvalue Problems

I Found in applications where goal is finding physicalparameters of a system based on known behavior orconstructing a system with physical parameters resulting ina desired dynamical behavior [3].

I Particle physicsI Molecular spectroscopyI Geophysics

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Applicationsof Inverse Eigenvalue Problems

I Found in applications where goal is finding physicalparameters of a system based on known behavior orconstructing a system with physical parameters resulting ina desired dynamical behavior [3].

I Particle physicsI Molecular spectroscopyI Geophysics

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

Applicationsof Inverse Eigenvalue Problems

I Found in applications where goal is finding physicalparameters of a system based on known behavior orconstructing a system with physical parameters resulting ina desired dynamical behavior [3].

I Particle physicsI Molecular spectroscopyI Geophysics

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

For Further Reading I

G. Strang.Introduction to Linear Algebra, Fourth Edition.Wesley-Cambridge, 2009.

Trustees of Princeton UniversityWordNet A Lexical Database for English, 2010http://wordnetweb.princeton.edu/perl/webwn?s=eigenvalue

Inverse Eigenvalue Problems College of the Redwoods

Introduction One Simple Algorithm Applications

For Further Reading II

M. Chu, G. Golub.Inverse Eigenvalue Problems: Theory and Applications.Department of Mathematics, North Carolina StateUniversity, 2001http://www4.ncsu.edu/~mtchu/Research/Lectures/Iep/preface.ps

K. Heuvers.Symmetric Matrices with Prescribed Eigenvalues andEigenvectorsMathematics Magazine, Vol. 55, No. 2. (Mar., 1982), pp.106-111

Inverse Eigenvalue Problems College of the Redwoods