Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References Acyclic matrices with a small number of distinct eigenvalues Presenter: Xavier Mart´ ınez-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.
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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.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Outline
Basic terminology
Smith normal form technique
Acyclic matrices with few distinct eigenvalues
References
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}.
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}.
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}.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices
Definition
A matrix A is acyclic if it is symmetric and G (A) is a tree.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
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.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
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)
).
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
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)
).
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
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)
).
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
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 .
Theorem (Kim & Shader; 2009)
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.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
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 .
Theorem (Kim & Shader; 2009)
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.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
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 .
Theorem (Kim & Shader; 2009)
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.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
A well-known fact
Theorem
Let T be a tree and let A ∈ S(T ).Then q(A) ≥ diam(T ) + 1.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Another well-known fact
Proposition
If A ∈ S(Pn), then q(A) = n.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
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).
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
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).
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with exactly 2 distinct eigenvalues
Theorem (Nair & Shader; 2013)
Let A be an acyclic matrix. Thenq(A) = 2 ⇐⇒ G (A) = K2.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
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).
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
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).
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
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.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
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.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
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.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
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.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with exactly 3 distinct eigenvalues
Theorem (Nair & Shader; 2013)
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.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with exactly 3 distinct eigenvalues
Theorem (Nair & Shader; 2013)
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.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with exactly 3 distinct eigenvalues
Theorem (Nair & Shader; 2013)
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.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with exactly 3 distinct eigenvalues
Theorem (Nair & Shader; 2013)
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.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with exactly 3 distinct eigenvalues
Theorem (Nair & Shader; 2013)
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.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with exactly 3 distinct eigenvalues
Theorem (Nair & Shader; 2013)
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.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with exactly 3 distinct eigenvalues
Theorem (Nair & Shader; 2013)
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.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with exactly 3 distinct eigenvalues
Theorem (Nair & Shader; 2013)
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.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with exactly 3 distinct eigenvalues
Theorem (Nair & Shader; 2013)
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.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with exactly 3 distinct eigenvalues
Theorem (Nair & Shader; 2013)
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.
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
• 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.