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

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

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

Nonnegativity of the Nullspace of an M-matrix

The nullspace N(A) of a (reducible) M-matrix isnot necessarily spanned by nonnegative vectors

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

Frobenius Normal Form

Frobenius Normal Form of A

A11 0 0 · · · 0A21 A22 0 · · · 0A31 A32 A33 · · · 0...

Aq1 Aq2 · · · · · · Aqq

A11, A22, . . . , Aqq are square irreducible

Frobenius Normal Form

Frobenius Normal Form of A

A11 0 0 · · · 0A21 A22 0 · · · 0A31 A32 A33 · · · 0...

Aq1 Aq2 · · · · · · Aqq

A11, A22, . . . , Aqq are square irreducible

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

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

The Reduced Graph

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

The Reduced Graph

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)

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

Preferred Vectors

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

Preferred Vectors

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 . =

++++++

Preferred Vectors

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+++++

Preferred Vectors

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

Preferred Vectors

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+++

Preferred Vectors

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++

Preferred Nullvectors

Theorem

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

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)

0 0 0 00 0 0 0−1 −1 0 0−1 −1 0 0

λ(A) = (2, 2)

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

The Preferred Basis Theorem

The generalized nullspace E (A) of A is N(An)

The Preferred Basis Theorm

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 =∑

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

Level Basis

The level of a vector x is the maximal level of asingular vertex i in R(A) such that xi 6= 0.

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)

Level Basis

Jordan Basis

We do not necessarily have a nonnegative Jordanbasis

Jordan Basis

We do not necessarily have a nonnegative Jordanbasis

The nullspace is not necessarily spanned bynonnegative vectors

0 0 0 00 0 0 0−1 −1 0 0−1 −1 0 0

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

1 → ∗1 → ∗2 → ∗ ∗3 → ∗ ∗ ∗

↑ ↑ ↑4 2 1

(3, 2, 1, 1)∗ = (4, 2, 1)

1 → ∗1 → ∗2 → ∗ ∗3 → ∗ ∗ ∗

↑ ↑ ↑4 2 1

(3, 2, 1, 1)∗ = (4, 2, 1)

η(A) = j(A)∗

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.

Height Basis

A basis for E (A) is called a height basis if thenumber of basis elements of height j equals ηj(A)for all j

Height Basis

Every Jordan basis for A is a height basis

Height Basis

Every Jordan basis for A is a height basis

We do not necessarily have a nonnegative heightbasis

Nonnegative Height Basis

When do we have a nonnegative height basis?

Theorem

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.

When do we have λ(A) = η(A)?

Theorem

Theorem Schneider (1956)

Let A be an M-matrix. The following are equivalent(i) λ(A) = (t)(ii) η(A) = (t)

a 0 0 0−b 0 0 00 −d e 0−f 0 0 0

λ(A) = (2) = η(A)

Theorem

Let A be an M-matrix. The following are equivalent(i) λ(A) = (1, . . . , 1)(ii) η(A) = (1, . . . , 1)

a 0 0 0−b 0 0 0−c −d e 0−f −g −h 0

λ(A) = (1, 1) = η(A)

Question

Do we always have λ(A) = η(A)?

Question

NO !!!

Question

NO !!!

0 0 0 00 0 0 0−c −d 0 0−f −g 0 0

Question

NO !!!

0 0 0 00 0 0 0−c −d 0 0−f −g 0 0

Question

NO !!!

0 0 0 00 0 0 0−c −d 0 0−f −g 0 0

λ(A) = (2, 2)

Question

NO !!!

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)

How do λ(A) and η(A) relate?

Theorem

Theorem Richman-Schneider (1978)

For M-matrices we have λ(A) � η(A)

λ1 + . . . + λk ≤ η1 + . . . + ηk , k < t

λ1 + . . . + λt = η1 + . . . + ηt

λ1 + . . . + λk ≤ η1 + . . . + ηk , k < t

λ1 + . . . + λt = η1 + . . . + ηt

Question

What are all possible λ(A) for a given 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)?

Theorem

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

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)}

Theorem

{k(γ)}

Theorem Friedland-Hershkowitz (1988)

{k(γ)}

n(Aκ) ≥ Σqi=1n(Aki

{k(γ)}

λ(A) = λ(A) reordered in a non-increasing order

{k(γ)}

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)

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)

0 0 0 0 00 0 0 0 0∗ ∗ 0 0 0∗ ∗ ∗ 0 0∗ ∗ ∗ 0 0

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.

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)

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)

0 0 0 0 00 0 0 0 0∗ ∗ 0 0 0∗ ∗ ∗ 0 0∗ ∗ ∗ 0 0