Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues

Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References

Acyclic matrices with a small number of

distinct eigenvalues

Presenter: Xavier Mart́ınez-Rivera

Iowa State University

April 26, 2017

Reshmi Nair, Bryan Shader.Acyclic matrices with a small number of distinct eigenvalues.Linear Algebra and its Applications, 438 (2013), 4075–4089.


Outline

Basic terminology

Smith normal form technique

Acyclic matrices with few distinct eigenvalues

References


The graph of a symmetric matrix

Definition

Let A = [aij ] ∈ Rn×n be symmetric.The graph of A, denoted by G (A),whose vertex and edge set is V and E , respectively,is defined as follows:

1. V = {1, 2, . . . , n};2. For i 6= j in V , ij ∈ E ⇐⇒ aij 6= 0.

Definition

Let G be a simple, n-vertex graph. Then

S(G ) := {A ∈ Rn×n : AT = A and G (A) = G}.



Definition



Definition





Definition



Definition




Acyclic matrices

Definition

A matrix A is acyclic if it is symmetric and G (A) is a tree.


Smith normal form technique

In 2009, Kim and Shader introduced a technique for studyingthe multiplicities of the eigenvalues of an acyclic matrix Abased on the Smith normal form of the matrix xI − A.


Definition

Let B ∈ (R[x ])n×n.Then ∆k(B) is defined to bethe monic gcd of all k × k minors of B .

Notation: q(A) denotes the number of distinct eigenvalues ofa matrix A.

Theorem (Kim & Shader; 2009)

Let A ∈ Rn×n be symmetric and let B = xI − A.Then q(A) = n − degree

(∆n−1(B)

).


Definition





(∆n−1(B)

).


Definition





(∆n−1(B)

).


Notation: Given A ∈ Rn×n and X ⊂ {1, . . . , n},A(X ) denotes the submatrix of A resulting fromdeleting the rows and columns indexed by X .


Let T be a tree, let A ∈ S(T ) and let B = xI − A;let G1, . . . ,Gk be vertex-disjoint paths, whereV := V (G1) ∪ · · · ∪ V (Gn) contains t vertices.Then

1. ∆n−k(B) divides det(B(V )

);

2. degree(∆n−k(B)) ≤ n − t.






);







);



A well-known fact

Theorem

Let T be a tree and let A ∈ S(T ).Then q(A) ≥ diam(T ) + 1.


Another well-known fact

Proposition

If A ∈ S(Pn), then q(A) = n.


Notation: mA(λ) denotes the multiplicity of λ as aneigenvalue of A.

Lemma (Johnson, Duarte, Saiago; 2013)

Let T be a tree and let A ∈ S(T ).Then mA(λmax) = 1 = mA(λmin).


Notation: mA(λ) denotes the multiplicity of λ as aneigenvalue of A.

Lemma (Johnson, Duarte, Saiago; 2013)

Let T be a tree and let A ∈ S(T ).Then mA(λmax) = 1 = mA(λmin).


Acyclic matrices with exactly 2 distinct eigenvalues

Theorem (Nair & Shader; 2013)

Let A be an acyclic matrix. Thenq(A) = 2 ⇐⇒ G (A) = K2.


Type-I matrices

Definition (Nair & Shader; 2013)

Let n ≥ 4. The matrix A ∈ Rn×n is a type-I matrix if it issimilar via a permutation matrix to a symmetric matrix of theform

? ∗ ∗ · · · ∗∗ β∗ β...

. . .

∗ β

.

Observation (Nair & Shader; 2013)

The graph of a type-I matrix is a star (which is a tree).


Type-I matrices

Definition (Nair & Shader; 2013)

Let n ≥ 4. The matrix A ∈ Rn×n is a type-I matrix if it issimilar via a permutation matrix to a symmetric matrix of theform

? ∗ ∗ · · · ∗∗ β∗ β...

. . .

∗ β

.

Observation (Nair & Shader; 2013)

The graph of a type-I matrix is a star (which is a tree).


Remark (Nair & Shader; 2013)

Let A be an n × n type-I matrix. Then

1. q(A) = 3;

2. A has an eigenvalue of multiplicity n − 2.


Another well-known fact

Remark

Let A ∈ Rn×n be symmetric with λ as an eigenvalue.Then one of the following holds for all j ∈ {1, . . . , n}:

1. mA({j})(λ) = mA(λ)− 1;

2. mA({j})(λ) = mA(λ);

3. mA({j})(λ) = mA(λ) + 1.


The Parter-Wiener Theorem

Theorem (P-W Theorem)

Let T be a tree on n vertices.Let A ∈ S(T ), and let λ be an eigenvalue with mA(λ) ≥ 2.Then there is a vertex j such that mA({j})(λ) = mA(λ) + 1.

j is called a Parter vertex for λ.


The Parter-Wiener Theorem

Theorem (P-W Theorem)

Let T be a tree on n vertices.Let A ∈ S(T ), and let λ be an eigenvalue with mA(λ) ≥ 2.Then there is a vertex j such that mA({j})(λ) = mA(λ) + 1.

j is called a Parter vertex for λ.




Let A be an acyclic matrix. Thenq(A) = 3 ⇐⇒ A is a type-I matrix or G (A) = P3.

Proof:

• Let A be an n × n acyclic matrix.

• =⇒ A ∈ S(T ) for some tree T .

• Suppose q(A) = 3.

• =⇒ n ≥ 3.

• Let α < β < γ be the distinct eigenvalues of A.

• Recall that q(A) ≥ diam(T ) + 1.

• =⇒ diam(T ) ≤ 2.

• What trees on n ≥ 3 vertices have diameter 2?

• Answer: Stars and P3.





Proof:




• =⇒ n ≥ 3.



• =⇒ diam(T ) ≤ 2.







Proof:




• =⇒ n ≥ 3.



• =⇒ diam(T ) ≤ 2.







Proof:




• =⇒ n ≥ 3.



• =⇒ diam(T ) ≤ 2.







Proof:




• =⇒ n ≥ 3.



• =⇒ diam(T ) ≤ 2.







Proof:




• =⇒ n ≥ 3.



• =⇒ diam(T ) ≤ 2.







Proof:




• =⇒ n ≥ 3.



• =⇒ diam(T ) ≤ 2.







Proof:




• =⇒ n ≥ 3.



• =⇒ diam(T ) ≤ 2.







Proof:




• =⇒ n ≥ 3.



• =⇒ diam(T ) ≤ 2.







Proof:




• =⇒ n ≥ 3.



• =⇒ diam(T ) ≤ 2.




• If T = P3, then we are done.

• So, suppose T is a star on (n ≥ 4 vertices).

• Previous lemma =⇒ mA(α) = 1 = mA(γ).

• =⇒ mA(β) = n − 2.

• =⇒ mA(β) ≥ 2.

• The Parter-Wiener Theorem implies that T contains aParter vertex for β.

• Let j be a Parter vertex for β.

• =⇒ mA({j})(β) = mA(β) + 1 = (n − 2) + 1 = n − 1.

• Which vertex of the star could j be?

• Answer: it must be the central vertex. Why?





• =⇒ mA(β) = n − 2.

• =⇒ mA(β) ≥ 2.



• =⇒ mA({j})(β) = mA(β) + 1 = (n − 2) + 1 = n − 1.







• =⇒ mA(β) = n − 2.

• =⇒ mA(β) ≥ 2.



• =⇒ mA({j})(β) = mA(β) + 1 = (n − 2) + 1 = n − 1.







• =⇒ mA(β) = n − 2.

• =⇒ mA(β) ≥ 2.



• =⇒ mA({j})(β) = mA(β) + 1 = (n − 2) + 1 = n − 1.







• =⇒ mA(β) = n − 2.

• =⇒ mA(β) ≥ 2.



• =⇒ mA({j})(β) = mA(β) + 1 = (n − 2) + 1 = n − 1.







• =⇒ mA(β) = n − 2.

• =⇒ mA(β) ≥ 2.



• =⇒ mA({j})(β) = mA(β) + 1 = (n − 2) + 1 = n − 1.







• =⇒ mA(β) = n − 2.

• =⇒ mA(β) ≥ 2.



• =⇒ mA({j})(β) = mA(β) + 1 = (n − 2) + 1 = n − 1.







• =⇒ mA(β) = n − 2.

• =⇒ mA(β) ≥ 2.



• =⇒ mA({j})(β) = mA(β) + 1 = (n − 2) + 1 = n − 1.







• =⇒ mA(β) = n − 2.

• =⇒ mA(β) ≥ 2.



• =⇒ mA({j})(β) = mA(β) + 1 = (n − 2) + 1 = n − 1.







• =⇒ mA(β) = n − 2.

• =⇒ mA(β) ≥ 2.



• =⇒ mA({j})(β) = mA(β) + 1 = (n − 2) + 1 = n − 1.




• ? ∗ ∗ · · · ∗∗ ?∗ ?...

. . .

∗ ?

.

• ? ∗ ∗ · · · ∗∗ β∗ β...

. . .

∗ β

.


• ? ∗ ∗ · · · ∗∗ ?∗ ?...

. . .

∗ ?

.•

? ∗ ∗ · · · ∗∗ β∗ β...

. . .

∗ β

.


Fin


I.-J. Kim, B. L. Shader.Smith Normal Form and acyclic matrices.J. Algebraic Combin. 29 (2009), 63–80.

R. Nair, B. L. Shader.Acyclic matrices with a small number of distincteigenvalues.Linear Algebra Appl. 438 (2013), 4075–4089.

C. R. Johnson, A. Leal Duarte, C. M. Saiago.The Parter Wiener theorem: refinement andgeneralization.SIAM J. Matrix Anal. Appl. 25 (2003), 352–361.

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Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues