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Spectral Properties of Nonnegative Matrices
Daniel Hershkowitz
Mathematics DepartmentTechnion - Israel Institute of Technology
Haifa 32000, Israel
December 1, 2008, Palo Alto
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Perron-Frobenius Theory
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Perron-Frobenius Theory
A is an n × n matrix
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Perron-Frobenius Theory
A is an n × n matrix
Perron-Frobenius (1912) Nonnegative Matrix Version
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Perron-Frobenius Theory
A is an n × n matrix
Perron-Frobenius (1912) Nonnegative Matrix Version
The largest absolute value ρ(A) of an eigenvalue ofa nonnegative matrix A is itself an eigenvalue of A,and it has an associated nonnegative eigenvector.Furthermore, if A is irreducible, ρ(A) is a simpleeigenvalue of A with an associated positiveeigenvector
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Perron-Frobenius Theory
A is a Z-matrix if A = rI − B where B isnonnegative entrywise
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Perron-Frobenius Theory
A is a Z-matrix if A = rI − B where B isnonnegative entrywise
A Z -matrix A = rI − B is an M-matrix if r ≥ ρ(B),where ρ(B) is the spectral radius of B
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Perron-Frobenius Theory
A is a Z-matrix if A = rI − B where B isnonnegative entrywise
A Z -matrix A = rI − B is an M-matrix if r ≥ ρ(B),where ρ(B) is the spectral radius of B
Perron-Frobenius (1912) M-Matrix Version
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Perron-Frobenius Theory
A is a Z-matrix if A = rI − B where B isnonnegative entrywise
A Z -matrix A = rI − B is an M-matrix if r ≥ ρ(B),where ρ(B) is the spectral radius of B
Perron-Frobenius (1912) M-Matrix Version
A singular M-matrix A has a nonnegative nullvector.Furthermore, if A is irreducible then 0 is a simpleeigenvalue of A with an associated positiveeigenvector
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nonnegativity of the Nullspace of an M-matrix
The nullspace N(A) of a (reducible) M-matrix isnot necessarily spanned by nonnegative vectors
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nonnegativity of the Nullspace of an M-matrix
The nullspace N(A) of a (reducible) M-matrix isnot necessarily spanned by nonnegative vectors
A =
0 0 0 00 0 0 0−1 −1 0 0−1 −1 0 0
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nonnegativity of the Nullspace of an M-matrix
The nullspace N(A) of a (reducible) M-matrix isnot necessarily spanned by nonnegative vectors
A =
0 0 0 00 0 0 0−1 −1 0 0−1 −1 0 0
Every nullvector (x1, x2, x3, x4)T satisfies x1 = −x2.
Since the nullity of A is 3, a basis for the nullspaceof A must contain a vector for which x1 = −x2 6= 0
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Frobenius Normal Form
Frobenius Normal Form of A
A =
A11 0 0 · · · 0A21 A22 0 · · · 0A31 A32 A33 · · · 0...
Aq1 Aq2 · · · · · · Aqq
A11, A22, . . . , Aqq are square irreducible
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Frobenius Normal Form
Frobenius Normal Form of A
A =
A11 0 0 · · · 0A21 A22 0 · · · 0A31 A32 A33 · · · 0...
Aq1 Aq2 · · · · · · Aqq
A11, A22, . . . , Aqq are square irreducible
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Reduced Graph
The reduced graph R(A):vertices: 1, . . . , q
arc i → j iff Aij 6= 0
A vertex i in R(A) is singular if Aii is singular
For a singular vertex i in R(A) we define level(i) asthe maximal number of singular vertices on a pathin R(A) that terminates at i
λk = number of singular vertices of R(A) of level k
λ(A) = (λ1, . . . , λt) = the level characteristic of ADaniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Reduced Graph
The reduced graph R(A):vertices: 1, . . . , q
arc i → j iff Aij 6= 0
A vertex i in R(A) is singular if Aii is singular
For a singular vertex i in R(A) we define level(i) asthe maximal number of singular vertices on a pathin R(A) that terminates at i
λk = number of singular vertices of R(A) of level k
λ(A) = (λ1, . . . , λt) = the level characteristic of ADaniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Reduced Graph
The reduced graph R(A):vertices: 1, . . . , q
arc i → j iff Aij 6= 0
A vertex i in R(A) is singular if Aii is singular
For a singular vertex i in R(A) we define level(i) asthe maximal number of singular vertices on a pathin R(A) that terminates at i
λk = number of singular vertices of R(A) of level k
λ(A) = (λ1, . . . , λt) = the level characteristic of ADaniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Reduced Graph
The reduced graph R(A):vertices: 1, . . . , q
arc i → j iff Aij 6= 0
A vertex i in R(A) is singular if Aii is singular
For a singular vertex i in R(A) we define level(i) asthe maximal number of singular vertices on a pathin R(A) that terminates at i
λk = number of singular vertices of R(A) of level k
λ(A) = (λ1, . . . , λt) = the level characteristic of ADaniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Reduced Graph
The reduced graph R(A):vertices: 1, . . . , q
arc i → j iff Aij 6= 0
A vertex i in R(A) is singular if Aii is singular
For a singular vertex i in R(A) we define level(i) asthe maximal number of singular vertices on a pathin R(A) that terminates at i
λk = number of singular vertices of R(A) of level k
λ(A) = (λ1, . . . , λt) = the level characteristic of ADaniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Reduced Graph
A =
0 0 0 0 0 0−5 3 0 0 0 00 −2 2 0 0 0−1 −1 0 0 0 0−1 0 0 −1 3 −2−2 0 0 0 −3 2
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Reduced Graph
A =
0 0 0 0 0 0−5 3 0 0 0 00 −2 2 0 0 0−1 −1 0 0 0 0−1 0 0 −1 3 −2−2 0 0 0 −3 2
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Reduced Graph
A =
0 0 0 0 0 0−5 3 0 0 0 00 −2 2 0 0 0−1 −1 0 0 0 0−1 0 0 −1 3 −2−2 0 0 0 −3 2
λ(A) = (1, 1, 1)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Preferred Vectors
Let x be vector in IRn partitioned conformably with
the Frobenius normal form A, and let i be a vertexin R(A). We say that x is an i -preferred vector(with respect to A) if
{
xj > 0, there is a path from j to i in R(A)xj = 0, otherwise
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Preferred Vectors
A =
0 0 0 0 0 0−5 3 0 0 0 00 −2 2 0 0 0−1 −1 0 0 0 0−1 0 0 −1 3 −2−2 0 0 0 −3 2
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Preferred Vectors
A =
0 0 0 0 0 0−5 3 0 0 0 00 −2 2 0 0 0−1 −1 0 0 0 0−1 0 0 −1 3 −2−2 0 0 0 −3 2
1−pref . =
++++++
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Preferred Vectors
A =
0 0 0 0 0 0−5 3 0 0 0 00 −2 2 0 0 0−1 −1 0 0 0 0−1 0 0 −1 3 −2−2 0 0 0 −3 2
1−pref . =
++++++
2−pref . =
0+++++
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Preferred Vectors
A =
0 0 0 0 0 0−5 3 0 0 0 00 −2 2 0 0 0−1 −1 0 0 0 0−1 0 0 −1 3 −2−2 0 0 0 −3 2
1−pref . =
++++++
2−pref . =
0+++++
3−pref . =
00+000
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Preferred Vectors
A =
0 0 0 0 0 0−5 3 0 0 0 00 −2 2 0 0 0−1 −1 0 0 0 0−1 0 0 −1 3 −2−2 0 0 0 −3 2
1−pref . =
++++++
2−pref . =
0+++++
3−pref . =
00+000
4−pref . =
000+++
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Preferred Vectors
A =
0 0 0 0 0 0−5 3 0 0 0 00 −2 2 0 0 0−1 −1 0 0 0 0−1 0 0 −1 3 −2−2 0 0 0 −3 2
1−pref . =
++++++
2−pref . =
0+++++
3−pref . =
00+000
4−pref . =
000+++
5−pref . =
0000++
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Preferred Nullvectors
Theorem
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Preferred Nullvectors
Theorem Schneider - thesis (1952)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Preferred Nullvectors
Theorem Schneider - thesis (1952)
Let A be a singular M-matrix. For every level 1singular vertex i in R(A) there exists a unique (upto scalar multiples) nullvector x i for A which isi -preferred
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Preferred Nullvectors
Theorem Schneider - thesis (1952)
Let A be a singular M-matrix. For every level 1singular vertex i in R(A) there exists a unique (upto scalar multiples) nullvector x i for A which isi -preferred
Theorem
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Preferred Nullvectors
Theorem Schneider - thesis (1952)
Let A be a singular M-matrix. For every level 1singular vertex i in R(A) there exists a unique (upto scalar multiples) nullvector x i for A which isi -preferred
Theorem Carlson (1963)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Preferred Nullvectors
Theorem Schneider - thesis (1952)
Let A be a singular M-matrix. For every level 1singular vertex i in R(A) there exists a unique (upto scalar multiples) nullvector x i for A which isi -preferred
Theorem Carlson (1963)
Let A be a singular M-matrix. Every nonnegativenullvector for A is a linear combination withnonnegative coefficients of the i -preferrednullvectors that correspond to the level 1 singularvertices i in R(A)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Preferred Nullvectors
The nullspace N(A) of a (reducible) M-matrix isnot necessarily spanned by nonnegative vectors
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Preferred Nullvectors
The nullspace N(A) of a (reducible) M-matrix isnot necessarily spanned by nonnegative vectors
A =
0 0 0 00 0 0 0−1 −1 0 0−1 −1 0 0
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Preferred Nullvectors
The nullspace N(A) of a (reducible) M-matrix isnot necessarily spanned by nonnegative vectors
A =
0 0 0 00 0 0 0−1 −1 0 0−1 −1 0 0
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Preferred Nullvectors
The nullspace N(A) of a (reducible) M-matrix isnot necessarily spanned by nonnegative vectors
A =
0 0 0 00 0 0 0−1 −1 0 0−1 −1 0 0
λ(A) = (2, 2)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Preferred Nullvectors
The nullspace N(A) of a (reducible) M-matrix isnot necessarily spanned by nonnegative vectors
A =
0 0 0 00 0 0 0−1 −1 0 0−1 −1 0 0
λ(A) = (2, 2)
Every nullvector (x1, x2, x3, x4)T satisfies x1 = −x2.
Since the nullity of A is 3, a basis for the nullspaceof A must contain a vector for which x1 = −x2 6= 0
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Preferred Basis Theorem
The generalized nullspace E (A) of A is N(An)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Preferred Basis Theorem
The generalized nullspace E (A) of A is N(An)
The Preferred Basis Theorm
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Preferred Basis Theorem
The generalized nullspace E (A) of A is N(An)
The Preferred Basis TheormSchneider (1956), Rothblum (1975), Richman-Schneider (1978)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Preferred Basis Theorem
The generalized nullspace E (A) of A is N(An)
The Preferred Basis TheormSchneider (1956), Rothblum (1975), Richman-Schneider (1978)
Let A be a singular M-matrix, and let S be the setof singular vertices in R(A). Then there exists abasis for E (A) consisting of i -preferred vectors,i ∈ S .
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Preferred Basis Theorem
The generalized nullspace E (A) of A is N(An)
The Preferred Basis TheormSchneider (1956), Rothblum (1975), Richman-Schneider (1978)
Let A be a singular M-matrix, and let S be the setof singular vertices in R(A). Then there exists abasis for E (A) consisting of i -preferred vectors,i ∈ S .
−Ax i =∑
k∈S
cikxk, i ∈ S
where the coefficients cik satisfy{
cik > 0, k 6= i and there is a path from k to i in R(A)cik = 0, otherwise
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Level Basis
The level of a vector x is the maximal level of asingular vertex i in R(A) such that xi 6= 0.
λk = number of singular vertices of R(A) of level k
λ(A) = (λ1, . . . , λt) = the level characteristic of A
A basis for E (A) in which number of basis elementsof level j equals λj all j is called a level basis
The Preferred Basis Theorem states that for asingular M-matrix A there exists a nonnegative levelbasis for E (A)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Level Basis
The level of a vector x is the maximal level of asingular vertex i in R(A) such that xi 6= 0.
λk = number of singular vertices of R(A) of level k
λ(A) = (λ1, . . . , λt) = the level characteristic of A
A basis for E (A) in which number of basis elementsof level j equals λj all j is called a level basis
The Preferred Basis Theorem states that for asingular M-matrix A there exists a nonnegative levelbasis for E (A)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Level Basis
The level of a vector x is the maximal level of asingular vertex i in R(A) such that xi 6= 0.
λk = number of singular vertices of R(A) of level k
λ(A) = (λ1, . . . , λt) = the level characteristic of A
A basis for E (A) in which number of basis elementsof level j equals λj all j is called a level basis
The Preferred Basis Theorem states that for asingular M-matrix A there exists a nonnegative levelbasis for E (A)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Level Basis
The level of a vector x is the maximal level of asingular vertex i in R(A) such that xi 6= 0.
λk = number of singular vertices of R(A) of level k
λ(A) = (λ1, . . . , λt) = the level characteristic of A
A basis for E (A) in which number of basis elementsof level j equals λj all j is called a level basis
The Preferred Basis Theorem states that for asingular M-matrix A there exists a nonnegative levelbasis for E (A)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Level Basis
The level of a vector x is the maximal level of asingular vertex i in R(A) such that xi 6= 0.
λk = number of singular vertices of R(A) of level k
λ(A) = (λ1, . . . , λt) = the level characteristic of A
A basis for E (A) in which number of basis elementsof level j equals λj all j is called a level basis
The Preferred Basis Theorem states that for asingular M-matrix A there exists a nonnegative levelbasis for E (A)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Jordan Basis
We do not necessarily have a nonnegative Jordanbasis
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Jordan Basis
We do not necessarily have a nonnegative Jordanbasis
The nullspace is not necessarily spanned bynonnegative vectors
A =
0 0 0 00 0 0 0−1 −1 0 0−1 −1 0 0
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Height Characteristic
Segre characteristic of A = the (non-increasing) sequence j(A) of sizes ofJordan blocks associated with the eigenvalue 0
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Height Characteristic
Segre characteristic of A = the (non-increasing) sequence j(A) of sizes ofJordan blocks associated with the eigenvalue 0
The height characteristic of A is the sequence η(A) of differencesn(Ai ) − n(Ai−1) (n(A0) = 0)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Height Characteristic
Segre characteristic of A = the (non-increasing) sequence j(A) of sizes ofJordan blocks associated with the eigenvalue 0
The height characteristic of A is the sequence η(A) of differencesn(Ai ) − n(Ai−1) (n(A0) = 0)
1 → ∗1 → ∗2 → ∗ ∗3 → ∗ ∗ ∗
↑ ↑ ↑4 2 1
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Height Characteristic
Segre characteristic of A = the (non-increasing) sequence j(A) of sizes ofJordan blocks associated with the eigenvalue 0
The height characteristic of A is the sequence η(A) of differencesn(Ai ) − n(Ai−1) (n(A0) = 0)
1 → ∗1 → ∗2 → ∗ ∗3 → ∗ ∗ ∗
↑ ↑ ↑4 2 1
(3, 2, 1, 1)∗ = (4, 2, 1)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
The Height Characteristic
Segre characteristic of A = the (non-increasing) sequence j(A) of sizes ofJordan blocks associated with the eigenvalue 0
The height characteristic of A is the sequence η(A) of differencesn(Ai ) − n(Ai−1) (n(A0) = 0)
1 → ∗1 → ∗2 → ∗ ∗3 → ∗ ∗ ∗
↑ ↑ ↑4 2 1
(3, 2, 1, 1)∗ = (4, 2, 1)
η(A) = j(A)∗
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Height Basis
For a vector x in E (A) we define the height of x,denoted by height(x), to be the minimalnonnegative integer k such that Akx = 0.
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Height Basis
For a vector x in E (A) we define the height of x,denoted by height(x), to be the minimalnonnegative integer k such that Akx = 0.
A basis for E (A) is called a height basis if thenumber of basis elements of height j equals ηj(A)for all j
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Height Basis
For a vector x in E (A) we define the height of x,denoted by height(x), to be the minimalnonnegative integer k such that Akx = 0.
A basis for E (A) is called a height basis if thenumber of basis elements of height j equals ηj(A)for all j
Every Jordan basis for A is a height basis
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Height Basis
For a vector x in E (A) we define the height of x,denoted by height(x), to be the minimalnonnegative integer k such that Akx = 0.
A basis for E (A) is called a height basis if thenumber of basis elements of height j equals ηj(A)for all j
Every Jordan basis for A is a height basis
We do not necessarily have a nonnegative heightbasis
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nonnegative Height Basis
When do we have a nonnegative height basis?
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nonnegative Height Basis
When do we have a nonnegative height basis?
Theorem
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nonnegative Height Basis
When do we have a nonnegative height basis?
TheoremCarlson (1956), Richman-Schneider (1978), Hershkowitz-Schneider (1989)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nonnegative Height Basis
When do we have a nonnegative height basis?
TheoremCarlson (1956), Richman-Schneider (1978), Hershkowitz-Schneider (1989)
Let A be an M-matrix. The .following areequivalent:(i) λ(A) = η(A).(ii) For all x ∈ E (A) we have height(x) = level(x).(iii) Every height basis for E (A) is a level basis.(v) Every level basis for for E (A) is a height basis.(vi) There exists a nonnegative height basis forE (A).(vii) There exists a nonnegative Jordan basis for−A.
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
When do we have λ(A) = η(A)?
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
When do we have λ(A) = η(A)?
Theorem
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
When do we have λ(A) = η(A)?
Theorem Schneider (1956)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
When do we have λ(A) = η(A)?
Theorem Schneider (1956)
Let A be an M-matrix. The following are equivalent(i) λ(A) = (t)(ii) η(A) = (t)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
When do we have λ(A) = η(A)?
Theorem Schneider (1956)
Let A be an M-matrix. The following are equivalent(i) λ(A) = (t)(ii) η(A) = (t)
A =
a 0 0 0−b 0 0 00 −d e 0−f 0 0 0
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
When do we have λ(A) = η(A)?
Theorem Schneider (1956)
Let A be an M-matrix. The following are equivalent(i) λ(A) = (t)(ii) η(A) = (t)
A =
a 0 0 0−b 0 0 00 −d e 0−f 0 0 0
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
When do we have λ(A) = η(A)?
Theorem Schneider (1956)
Let A be an M-matrix. The following are equivalent(i) λ(A) = (t)(ii) η(A) = (t)
A =
a 0 0 0−b 0 0 00 −d e 0−f 0 0 0
λ(A) = (2) = η(A)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
When do we have λ(A) = η(A)?
Theorem
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
When do we have λ(A) = η(A)?
Theorem Schneider (1956)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
When do we have λ(A) = η(A)?
Theorem Schneider (1956)
Let A be an M-matrix. The following are equivalent(i) λ(A) = (1, . . . , 1)(ii) η(A) = (1, . . . , 1)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
When do we have λ(A) = η(A)?
Theorem Schneider (1956)
Let A be an M-matrix. The following are equivalent(i) λ(A) = (1, . . . , 1)(ii) η(A) = (1, . . . , 1)
A =
a 0 0 0−b 0 0 0−c −d e 0−f −g −h 0
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
When do we have λ(A) = η(A)?
Theorem Schneider (1956)
Let A be an M-matrix. The following are equivalent(i) λ(A) = (1, . . . , 1)(ii) η(A) = (1, . . . , 1)
A =
a 0 0 0−b 0 0 0−c −d e 0−f −g −h 0
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
When do we have λ(A) = η(A)?
Theorem Schneider (1956)
Let A be an M-matrix. The following are equivalent(i) λ(A) = (1, . . . , 1)(ii) η(A) = (1, . . . , 1)
A =
a 0 0 0−b 0 0 0−c −d e 0−f −g −h 0
λ(A) = (1, 1) = η(A)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
When do we have λ(A) = η(A)?
Question
Do we always have λ(A) = η(A)?
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
When do we have λ(A) = η(A)?
Question
Do we always have λ(A) = η(A)?
NO !!!
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
When do we have λ(A) = η(A)?
Question
Do we always have λ(A) = η(A)?
NO !!!
A =
0 0 0 00 0 0 0−c −d 0 0−f −g 0 0
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
When do we have λ(A) = η(A)?
Question
Do we always have λ(A) = η(A)?
NO !!!
A =
0 0 0 00 0 0 0−c −d 0 0−f −g 0 0
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
When do we have λ(A) = η(A)?
Question
Do we always have λ(A) = η(A)?
NO !!!
A =
0 0 0 00 0 0 0−c −d 0 0−f −g 0 0
λ(A) = (2, 2)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
When do we have λ(A) = η(A)?
Question
Do we always have λ(A) = η(A)?
NO !!!
A =
0 0 0 00 0 0 0−c −d 0 0−f −g 0 0
λ(A) = (2, 2)
c = d = f = g =⇒ η(A) = (3, 1)c = d = f = 2g =⇒ η(A) = (2, 2)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
How do λ(A) and η(A) relate?
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
How do λ(A) and η(A) relate?
Theorem
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
How do λ(A) and η(A) relate?
Theorem Richman-Schneider (1978)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
How do λ(A) and η(A) relate?
Theorem Richman-Schneider (1978)
For M-matrices we have λ(A) � η(A)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
How do λ(A) and η(A) relate?
Theorem Richman-Schneider (1978)
For M-matrices we have λ(A) � η(A)
λ1 + . . . + λk ≤ η1 + . . . + ηk , k < t
λ1 + . . . + λt = η1 + . . . + ηt
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
How do λ(A) and η(A) relate?
Theorem Richman-Schneider (1978)
For M-matrices we have λ(A) � η(A)
λ1 + . . . + λk ≤ η1 + . . . + ηk , k < t
λ1 + . . . + λt = η1 + . . . + ηt
Question
What are all possible λ(A) for a given η(A)?
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
How do λ(A) and η(A) relate?
Theorem Richman-Schneider (1978)
For M-matrices we have λ(A) � η(A)
λ1 + . . . + λk ≤ η1 + . . . + ηk , k < t
λ1 + . . . + λt = η1 + . . . + ηt
Question
What are all possible λ(A) for a given η(A)?
Question
What are all possible η(A) for a given λ(A)?
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
How do λ(A) and η(A) relate?
Theorem
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
How do λ(A) and η(A) relate?
Theorem Hershkowitz-Schneider (1991)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
How do λ(A) and η(A) relate?
Theorem Hershkowitz-Schneider (1991)
Let λ and η be two sequencessuch that λ � η. Then thereexists a graph G such that forevery matrix A with G (A) = G
we have λ(A) = λ andη(A) = η
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Block Triangular Matrices
A is a general n × n matrix in a block triangularform with square diagonal blocks. The reducedgraph R(A) is defined as before
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Block Triangular Matrices
A is a general n × n matrix in a block triangularform with square diagonal blocks. The reducedgraph R(A) is defined as before
Assign (nonnegative integer) weights k1, . . . , kq tothe vertices of R(A)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Block Triangular Matrices
A is a general n × n matrix in a block triangularform with square diagonal blocks. The reducedgraph R(A) is defined as before
Assign (nonnegative integer) weights k1, . . . , kq tothe vertices of R(A)
For a path γ = (i1, . . . , is) in R(A) we definek(γ) = ki1 + . . . + kis
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Block Triangular Matrices
A is a general n × n matrix in a block triangularform with square diagonal blocks. The reducedgraph R(A) is defined as before
Assign (nonnegative integer) weights k1, . . . , kq tothe vertices of R(A)
For a path γ = (i1, . . . , is) in R(A) we definek(γ) = ki1 + . . . + kis
κ = maxpaths γ in R(A)
{k(γ)}
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Block Triangular Matrices
κ = maxpaths γ in R(A)
{k(γ)}
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Block Triangular Matrices
κ = maxpaths γ in R(A)
{k(γ)}
Theorem
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Block Triangular Matrices
κ = maxpaths γ in R(A)
{k(γ)}
Theorem Friedland-Hershkowitz (1988)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Block Triangular Matrices
κ = maxpaths γ in R(A)
{k(γ)}
Theorem Friedland-Hershkowitz (1988)
n(Aκ) ≥ Σqi=1n(Aki
ii )
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Block Triangular Matrices
κ = maxpaths γ in R(A)
{k(γ)}
Theorem Friedland-Hershkowitz (1988)
n(Aκ) ≥ Σqi=1n(Aki
ii )
λ(A) = λ(A) reordered in a non-increasing order
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Block Triangular Matrices
κ = maxpaths γ in R(A)
{k(γ)}
Theorem Friedland-Hershkowitz (1988)
n(Aκ) ≥ Σqi=1n(Aki
ii )
λ(A) = λ(A) reordered in a non-increasing order
Corollary
(i) λ1 + . . . + λk ≤ η1 + . . . + ηk , ∀k
(ii) If 0 is a simple eigenvalue of every singular Aii
then λ(A) � λ(A) � η(A)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nilpotent Triangular Matrices
The graph G (A) of an n × n matrix A:vertices: 1, . . . , n
arc i → j iff aij 6= 0
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nilpotent Triangular Matrices
The graph G (A) of an n × n matrix A:vertices: 1, . . . , n
arc i → j iff aij 6= 0
k-path = a set of vertices that can be covered by k
or fewer (vertex) disjoint paths
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nilpotent Triangular Matrices
The graph G (A) of an n × n matrix A:vertices: 1, . . . , n
arc i → j iff aij 6= 0
k-path = a set of vertices that can be covered by k
or fewer (vertex) disjoint paths
pk(A) = maximal cardinality of a k-path in G (A)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nilpotent Triangular Matrices
The graph G (A) of an n × n matrix A:vertices: 1, . . . , n
arc i → j iff aij 6= 0
k-path = a set of vertices that can be covered by k
or fewer (vertex) disjoint paths
pk(A) = maximal cardinality of a k-path in G (A)
π(A) = sequence of differences pk(A) − pk−1(A),(p0(A) = 0)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nilpotent Triangular Matrices
A =
0 0 0 0 00 0 0 0 0∗ ∗ 0 0 0∗ ∗ ∗ 0 0∗ ∗ ∗ 0 0
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nilpotent Triangular Matrices
A =
0 0 0 0 00 0 0 0 0∗ ∗ 0 0 0∗ ∗ ∗ 0 0∗ ∗ ∗ 0 0
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nilpotent Triangular Matrices
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nilpotent Triangular Matrices
Longest paths: (4,3,1), (4,3,2), (5,3,1), (5,3,2).Thus, p1(A) = 3. All vertices can be covered by two
paths, e.g. (4,3,1) and (5,2). Thus, p2(A) = 5.
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nilpotent Triangular Matrices
Longest paths: (4,3,1), (4,3,2), (5,3,1), (5,3,2).Thus, p1(A) = 3. All vertices can be covered by two
paths, e.g. (4,3,1) and (5,2). Thus, p2(A) = 5.
π(A) = (3, 2) = (2, 2, 1)∗ = λ(A)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nilpotent Triangular Matrices
A =
0 0 0 0 00 0 0 0 0∗ ∗ 0 0 0∗ ∗ ∗ 0 0∗ ∗ ∗ 0 0
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nilpotent Triangular Matrices
A =
0 0 0 0 00 0 0 0 0∗ ∗ 0 0 0∗ ∗ ∗ 0 0∗ ∗ ∗ 0 0
Possible height characteristics: (3,1,1), (2,2,1)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nilpotent Triangular Matrices
A =
0 0 0 0 00 0 0 0 0∗ ∗ 0 0 0∗ ∗ ∗ 0 0∗ ∗ ∗ 0 0
Possible height characteristics: (3,1,1), (2,2,1)
π(A)∗ = (2, 2, 1) � (3, 1, 1), (2, 2, 1)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nilpotent Triangular Matrices
A =
0 0 0 0 00 0 0 0 0∗ ∗ 0 0 0∗ ∗ ∗ 0 0∗ ∗ ∗ 0 0
Possible height characteristics: (3,1,1), (2,2,1)
π(A)∗ = (2, 2, 1) � (3, 1, 1), (2, 2, 1)
Question
Do we always have π(A)∗ � η(A)?
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nilpotent Triangular Matrices
A =
0 0 0 0 00 0 0 0 0∗ ∗ 0 0 0∗ ∗ ∗ 0 0∗ ∗ ∗ 0 0
Possible height characteristics: (3,1,1), (2,2,1)
π(A)∗ = (2, 2, 1) � (3, 1, 1), (2, 2, 1)
Question
Do we always have π(A)∗ � η(A)?
Theorem
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nilpotent Triangular Matrices
A =
0 0 0 0 00 0 0 0 0∗ ∗ 0 0 0∗ ∗ ∗ 0 0∗ ∗ ∗ 0 0
Possible height characteristics: (3,1,1), (2,2,1)
π(A)∗ = (2, 2, 1) � (3, 1, 1), (2, 2, 1)
Question
Do we always have π(A)∗ � η(A)?
Theorem Hershkowitz-Schneider (1993)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Nilpotent Triangular Matrices
A =
0 0 0 0 00 0 0 0 0∗ ∗ 0 0 0∗ ∗ ∗ 0 0∗ ∗ ∗ 0 0
Possible height characteristics: (3,1,1), (2,2,1)
π(A)∗ = (2, 2, 1) � (3, 1, 1), (2, 2, 1)
Question
Do we always have π(A)∗ � η(A)?
Theorem Hershkowitz-Schneider (1993)
For a nilpotent triangular matrix A we haveπ(A)∗ � η(A)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Triangular Matrices
k-path = a set of vertices that can be covered by k
or fewer (vertex) disjoint paths
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Triangular Matrices
k-path = a set of vertices that can be covered by k
or fewer (vertex) disjoint paths
BEFORE: pk(A) = maximal cardinality of a k-pathin G (A)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Triangular Matrices
k-path = a set of vertices that can be covered by k
or fewer (vertex) disjoint paths
BEFORE: pk(A) = maximal cardinality of a k-pathin G (A)
NEW: pk(A) = maximal number of loopless verticesin a k-path in G (A)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Triangular Matrices
k-path = a set of vertices that can be covered by k
or fewer (vertex) disjoint paths
BEFORE: pk(A) = maximal cardinality of a k-pathin G (A)
NEW: pk(A) = maximal number of loopless verticesin a k-path in G (A)
π(A) = sequence of differences pk(A) − pk−1(A),(p0(A) = 0)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Triangular Matrices
A =
0 0 0 0 00 0 0 0 0∗ ∗ ∗ 0 00 0 ∗ 0 00 0 ∗ 0 0
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Triangular Matrices
A =
0 0 0 0 00 0 0 0 0∗ ∗ ∗ 0 00 0 ∗ 0 00 0 ∗ 0 0
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Triangular Matrices
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Triangular Matrices
p1(A) = 2, p2(A) = 3, p3(A) = 4
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Triangular Matrices
p1(A) = 2, p2(A) = 3, p3(A) = 4
π(A) = (2, 1, 1)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Triangular Matrices
p1(A) = 2, p2(A) = 3, p3(A) = 4
π(A) = (2, 1, 1)
π(A)∗ = (3, 1)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Triangular Matrices
p1(A) = 2, p2(A) = 3, p3(A) = 4
π(A) = (2, 1, 1)
π(A)∗ = (3, 1)
A =
0 0 0 0 00 0 0 0 0∗ ∗ ∗ 0 00 0 ∗ 0 00 0 ∗ 0 0
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Triangular Matrices
p1(A) = 2, p2(A) = 3, p3(A) = 4
π(A) = (2, 1, 1)
π(A)∗ = (3, 1)
A =
0 0 0 0 00 0 0 0 0∗ ∗ ∗ 0 00 0 ∗ 0 00 0 ∗ 0 0
π(A)∗ = (3, 1) = η(A)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Triangular Matrices
Theorem
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Triangular Matrices
Theorem Hershkowitz-Schneider (1993)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Triangular Matrices
Theorem Hershkowitz-Schneider (1993)
For a triangular matrix A we have π(A)∗ � η(A)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Triangular Matrices
Theorem Hershkowitz-Schneider (1993)
For a triangular matrix A we have π(A)∗ � η(A)
Question
When do we have π(A)∗ = η(A)?
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Triangular Matrices
Theorem Hershkowitz-Schneider (1993)
For a triangular matrix A we have π(A)∗ � η(A)
Question
When do we have π(A)∗ = η(A)?
Let IF be a field with infinitely many elements
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Triangular Matrices
Theorem Hershkowitz-Schneider (1993)
For a triangular matrix A we have π(A)∗ � η(A)
Question
When do we have π(A)∗ = η(A)?
Let IF be a field with infinitely many elements
Theorem
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Triangular Matrices
Theorem Hershkowitz-Schneider (1993)
For a triangular matrix A we have π(A)∗ � η(A)
Question
When do we have π(A)∗ = η(A)?
Let IF be a field with infinitely many elements
Theorem Hershkowitz-Schneider (1993)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Triangular Matrices
Theorem Hershkowitz-Schneider (1993)
For a triangular matrix A we have π(A)∗ � η(A)
Question
When do we have π(A)∗ = η(A)?
Let IF be a field with infinitely many elements
Theorem Hershkowitz-Schneider (1993)
For a triangular matrix A we have π(A)∗ � η(A).Furthermore, the generic matrix A over IF withgraph G (A) satisfies π(A)∗ = η(A)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Matrices
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Matrices
A path (i1, . . . , im) is closable if (im, i1) is an arc
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Matrices
A path (i1, . . . , im) is closable if (im, i1) is an arc
BEFORE: k-path = a set of vertices that can becovered by k or fewer (vertex) disjoint paths
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Matrices
A path (i1, . . . , im) is closable if (im, i1) is an arc
BEFORE: k-path = a set of vertices that can becovered by k or fewer (vertex) disjoint paths
NEW: k-path = a set of vertices that can becovered by disjoint paths, where the number of thenon-closable paths in this cover does not exceed k
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Matrices
A path (i1, . . . , im) is closable if (im, i1) is an arc
BEFORE: k-path = a set of vertices that can becovered by k or fewer (vertex) disjoint paths
NEW: k-path = a set of vertices that can becovered by disjoint paths, where the number of thenon-closable paths in this cover does not exceed k
pk(A) = maximal cardinality of a k-path in G (A)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Matrices
A path (i1, . . . , im) is closable if (im, i1) is an arc
BEFORE: k-path = a set of vertices that can becovered by k or fewer (vertex) disjoint paths
NEW: k-path = a set of vertices that can becovered by disjoint paths, where the number of thenon-closable paths in this cover does not exceed k
pk(A) = maximal cardinality of a k-path in G (A)
π(A) = sequence of differences pk(A) − pk−1(A),(p0(A) is the maximal number of vertices that can
be covered by disjoint closable paths)Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Matrices
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Matrices
p0(A) = 4 (1, 4, 5 and 7)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Matrices
p0(A) = 4 (1, 4, 5 and 7)
p1(A) = 6 (e.g. the closable (4,7,5) and (1) and the non-closable (3,2).Or: the closable (1) and the non-closable (4,7,5,3,2))
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Matrices
p0(A) = 4 (1, 4, 5 and 7)
p1(A) = 6 (e.g. the closable (4,7,5) and (1) and the non-closable (3,2).Or: the closable (1) and the non-closable (4,7,5,3,2))
p2(A) = 7
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Matrices
p0(A) = 4 (1, 4, 5 and 7)
p1(A) = 6 (e.g. the closable (4,7,5) and (1) and the non-closable (3,2).Or: the closable (1) and the non-closable (4,7,5,3,2))
p2(A) = 7
π(A) = (2, 1) = π(A)∗
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Matrices
A =
∗ 0 0 0 0 0 00 0 0 0 0 0 0∗ ∗ 0 0 0 0 00 0 0 0 0 0 ∗0 0 ∗ ∗ 0 0 00 0 0 0 ∗ 0 00 0 0 0 ∗ 0 0
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Matrices
A =
∗ 0 0 0 0 0 00 0 0 0 0 0 0∗ ∗ 0 0 0 0 00 0 0 0 0 0 ∗0 0 ∗ ∗ 0 0 00 0 0 0 ∗ 0 00 0 0 0 ∗ 0 0
A2 =
∗ 0 0 0 0 0 00 0 0 0 0 0 0∗ 0 0 0 0 0 00 0 0 0 ∗ 0 0∗ ∗ 0 0 0 0 ∗0 0 ∗ ∗ 0 0 00 0 ∗ ∗ 0 0 0
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Matrices
A =
∗ 0 0 0 0 0 00 0 0 0 0 0 0∗ ∗ 0 0 0 0 00 0 0 0 0 0 ∗0 0 ∗ ∗ 0 0 00 0 0 0 ∗ 0 00 0 0 0 ∗ 0 0
A2 =
∗ 0 0 0 0 0 00 0 0 0 0 0 0∗ 0 0 0 0 0 00 0 0 0 ∗ 0 0∗ ∗ 0 0 0 0 ∗0 0 ∗ ∗ 0 0 00 0 ∗ ∗ 0 0 0
η(A) = (2, 1)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Matrices
A =
∗ 0 0 0 0 0 00 0 0 0 0 0 0∗ ∗ 0 0 0 0 00 0 0 0 0 0 ∗0 0 ∗ ∗ 0 0 00 0 0 0 ∗ 0 00 0 0 0 ∗ 0 0
A2 =
∗ 0 0 0 0 0 00 0 0 0 0 0 0∗ 0 0 0 0 0 00 0 0 0 ∗ 0 0∗ ∗ 0 0 0 0 ∗0 0 ∗ ∗ 0 0 00 0 ∗ ∗ 0 0 0
η(A) = (2, 1)
Let IF be a field with infinitely many elements
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Matrices
A =
∗ 0 0 0 0 0 00 0 0 0 0 0 0∗ ∗ 0 0 0 0 00 0 0 0 0 0 ∗0 0 ∗ ∗ 0 0 00 0 0 0 ∗ 0 00 0 0 0 ∗ 0 0
A2 =
∗ 0 0 0 0 0 00 0 0 0 0 0 0∗ 0 0 0 0 0 00 0 0 0 ∗ 0 0∗ ∗ 0 0 0 0 ∗0 0 ∗ ∗ 0 0 00 0 ∗ ∗ 0 0 0
η(A) = (2, 1)
Let IF be a field with infinitely many elements
Theorem
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Matrices
A =
∗ 0 0 0 0 0 00 0 0 0 0 0 0∗ ∗ 0 0 0 0 00 0 0 0 0 0 ∗0 0 ∗ ∗ 0 0 00 0 0 0 ∗ 0 00 0 0 0 ∗ 0 0
A2 =
∗ 0 0 0 0 0 00 0 0 0 0 0 0∗ 0 0 0 0 0 00 0 0 0 ∗ 0 0∗ ∗ 0 0 0 0 ∗0 0 ∗ ∗ 0 0 00 0 ∗ ∗ 0 0 0
η(A) = (2, 1)
Let IF be a field with infinitely many elements
Theorem Hershkowitz (1993)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
General Matrices
A =
∗ 0 0 0 0 0 00 0 0 0 0 0 0∗ ∗ 0 0 0 0 00 0 0 0 0 0 ∗0 0 ∗ ∗ 0 0 00 0 0 0 ∗ 0 00 0 0 0 ∗ 0 0
A2 =
∗ 0 0 0 0 0 00 0 0 0 0 0 0∗ 0 0 0 0 0 00 0 0 0 ∗ 0 0∗ ∗ 0 0 0 0 ∗0 0 ∗ ∗ 0 0 00 0 ∗ ∗ 0 0 0
η(A) = (2, 1)
Let IF be a field with infinitely many elements
Theorem Hershkowitz (1993)
For every square matrix A we have π(A)∗ ≪ η(A).Furthermore, the generic matrix A over IF with graph G (A)satisfies π(A)∗ = η(A)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Back to Frobenius Normal Form
The Segre characteristic j(A) = (j1, . . . , jt)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Back to Frobenius Normal Form
The Segre characteristic j(A) = (j1, . . . , jt)
GJ(A) = the graph consisting of t disjoint paths of looplessvertices and of lengths j1, . . . , jt , and of rank(A) singletons
with loops
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Back to Frobenius Normal Form
The Segre characteristic j(A) = (j1, . . . , jt)
GJ(A) = the graph consisting of t disjoint paths of looplessvertices and of lengths j1, . . . , jt , and of rank(A) singletons
with loops
A =
A11 0 0 · · · 0A21 A22 0 · · · 0A31 A32 A33 · · · 0...
Aq1 Aq2 · · · · · · Aqq
, A11, . . . , Aqq irreducible
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Back to Frobenius Normal Form
The Segre characteristic j(A) = (j1, . . . , jt)
GJ(A) = the graph consisting of t disjoint paths of looplessvertices and of lengths j1, . . . , jt , and of rank(A) singletons
with loops
A =
A11 0 0 · · · 0A21 A22 0 · · · 0A31 A32 A33 · · · 0...
Aq1 Aq2 · · · · · · Aqq
, A11, . . . , Aqq irreducible
RJ(A) = the graph obtained by taking q disjoint graphsGJ(A11), . . . , GJ(Aqq), and adding arcs from every vertex ofGJ(Aii) to every vertex of GJ(Ajj) whenever Aij 6= 0, i 6= j
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Back to Frobenius Normal Form
A =
1 1 1 0 0 0 01 1 1 0 0 0 0−2 −2 −2 0 0 0 00 0 0 1 1 0 00 0 0 1 1 0 00 0 0 0 0 1 20 1 0 0 2 3 4
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Back to Frobenius Normal Form
A =
1 1 1 0 0 0 01 1 1 0 0 0 0−2 −2 −2 0 0 0 00 0 0 1 1 0 00 0 0 1 1 0 00 0 0 0 0 1 20 1 0 0 2 3 4
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Back to Frobenius Normal Form
Theorem
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Back to Frobenius Normal Form
Theorem Hershkowitz (1993)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices
Back to Frobenius Normal Form
Theorem Hershkowitz (1993)
For every square matrix A wehave π(RJ(A))∗ � η(A).Furthermore, the genericmatrix A over IF satisfiesπ(RJ(A))∗ = η(A)
Daniel Hershkowitz Spectral Properties of Nonnegative Matrices