Spectral Properties of Nonnegative Matrices · Daniel Hershkowitz Spectral Properties of...

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




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





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




A =

0 0 0 00 0 0 0−1 −1 0 0−1 −1 0 0




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


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


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


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)


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

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


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 . =

++++++


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+++++


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


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+++


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++


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)





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)




A =

0 0 0 00 0 0 0−1 −1 0 0−1 −1 0 0




A =

0 0 0 00 0 0 0−1 −1 0 0−1 −1 0 0

λ(A) = (2, 2)




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


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 .





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


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)


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

A =

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)




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 =

a 0 0 0−b 0 0 00 −d e 0−f 0 0 0





A =

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 =

a 0 0 0−b 0 0 0−c −d e 0−f −g −h 0





A =

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 !!!

A =

0 0 0 00 0 0 0−c −d 0 0−f −g 0 0



Question


NO !!!

A =

0 0 0 00 0 0 0−c −d 0 0−f −g 0 0

λ(A) = (2, 2)



Question


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)


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





For a path γ = (i1, . . . , is) in R(A) we definek(γ) = ki1 + . . . + kis

κ = maxpaths γ in R(A)

{k(γ)}




{k(γ)}




{k(γ)}

Theorem




{k(γ)}

Theorem Friedland-Hershkowitz (1988)




{k(γ)}


n(Aκ) ≥ Σqi=1n(Aki

ii )




{k(γ)}



ii )

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




{k(γ)}



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)


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)



A =

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)



A =

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



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)



A =

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


π(A)∗ = (2, 2, 1) � (3, 1, 1), (2, 2, 1)



A =

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


π(A)∗ = (2, 2, 1) � (3, 1, 1), (2, 2, 1)

Question

Do we always have π(A)∗ � η(A)?



A =

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


π(A)∗ = (2, 2, 1) � (3, 1, 1), (2, 2, 1)

Question


Theorem



A =

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


π(A)∗ = (2, 2, 1) � (3, 1, 1), (2, 2, 1)

Question





A =

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


π(A)∗ = (2, 2, 1) � (3, 1, 1), (2, 2, 1)

Question



For a nilpotent triangular matrix A we haveπ(A)∗ � η(A)


Triangular Matrices




Triangular Matrices



BEFORE: pk(A) = maximal cardinality of a k-pathin G (A)


Triangular Matrices




NEW: pk(A) = maximal number of loopless verticesin a k-path in G (A)


Triangular Matrices




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)


Triangular Matrices

A =

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


Triangular Matrices


Triangular Matrices

p1(A) = 2, p2(A) = 3, p3(A) = 4


Triangular Matrices

p1(A) = 2, p2(A) = 3, p3(A) = 4

π(A) = (2, 1, 1)


Triangular Matrices

p1(A) = 2, p2(A) = 3, p3(A) = 4

π(A) = (2, 1, 1)

π(A)∗ = (3, 1)


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


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)


Triangular Matrices

Theorem


Triangular Matrices



Triangular Matrices


For a triangular matrix A we have π(A)∗ � η(A)


Triangular Matrices



Question

When do we have π(A)∗ = η(A)?


Triangular Matrices



Question


Let IF be a field with infinitely many elements


Triangular Matrices



Question



Theorem


Triangular Matrices



Question





Triangular Matrices



Question




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)


General Matrices


General Matrices

A path (i1, . . . , im) is closable if (im, i1) is an arc


General Matrices


BEFORE: k-path = a set of vertices that can becovered by k or fewer (vertex) disjoint paths


General Matrices



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


General Matrices






General Matrices





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


General Matrices

p0(A) = 4 (1, 4, 5 and 7)


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))


General Matrices

p0(A) = 4 (1, 4, 5 and 7)


p2(A) = 7


General Matrices

p0(A) = 4 (1, 4, 5 and 7)


p2(A) = 7

π(A) = (2, 1) = π(A)∗


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


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


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)


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)


Theorem


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)


Theorem Hershkowitz (1993)


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)



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)


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





with loops

A =

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

Aq1 Aq2 · · · · · · Aqq

, A11, . . . , Aqq irreducible





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



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



Theorem




For every square matrix A wehave π(RJ(A))∗ � η(A).Furthermore, the genericmatrix A over IF satisfiesπ(RJ(A))∗ = η(A)


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