Kenneth Kuttler

6.11. CAUCHY’S INTERLACING THEOREM FOR EIGENVALUES 185

Hint: First note that there exists a vector a such that Aa is not a multiple of a. Thenconsider

B =(

a Aa)−1

a Aa)

Show B has a zero on the main diagonal.

49. ↑ Let A be a complex n×n matrix which has trace equal to 0. Show that A is similarto a matrix which has all zeros on the main diagonal. Hint: Use Problem 30 onPage 128 to argue that you can say that a given matrix is similar to one which hasthe diagonal entries permuted in any order desired. Then use the above problem andblock multiplication to show that if the A has k nonzero entries, then it is similar toa matrix which has k− 1 nonzero entries. Finally, when A is similar to one which hasat most one nonzero entry, this one must also be zero because of the condition on thetrace.

50. ↑An n × n matrix X is a comutator if there are n × n matrices A,B such that X =AB − BA. Show that the trace of any comutator is 0. Next show that if a complexmatrix X has trace equal to 0, then it is in fact a comutator. Hint: Use the aboveproblem to show that it suffices to consider X having all zero entries on the maindiagonal. Then define

A =

1 0

2. . .

0 n

 , Bij =

{Xij

i−j if i ̸= j

0 if i = j

6.11 Cauchy’s Interlacing Theorem for Eigenvalues

Recall that every Hermitian matrix has all real eigenvalues. The Cauchy interlacing theoremcompares the location of the eigenvalues of a Hermitian matrix with the eigenvalues of aprincipal submatrix. It is an extremely interesting theorem.

Theorem 6.11.1 Let A be a Hermitian n× n matrix and let

A =

(a y∗

y B

)where B is (n− 1) × (n− 1) . Let the eigenvalues of B be µ1 ≤ µ2 ≤ · · · ≤ µn−1. Then ifthe eigenvalues of A are λ1 ≤ λ2 ≤ · · · ≤ λn, it follows that λ1 ≤ µ1 ≤ λ2 ≤ µ2 ≤ · · · ≤µn−1 ≤ λn.

Proof: First note that B is Hermitian because

A∗ =

(a y∗

y B∗

)= A =

(a y∗

y B

)It is easiest to consider the case where strict inequality holds for the eigenvalues for B sofirst is an outline of reducing to this case.

There exists U unitary, depending on B such that U∗BU = D where

D =

µ1 0

. . .

0 µn−1



6.11. CAUCHY’S INTERLACING THEOREM FOR EIGENVALUES 185Hint: First note that there exists a vector a such that Aa is not a multiple of a. Thenconsider-1B=(a Aa ) A(a Aa )Show B has a zero on the main diagonal.49. + Let A be a complex n X n matrix which has trace equal to 0. Show that A is similarto a matrix which has all zeros on the main diagonal. Hint: Use Problem 30 onPage 128 to argue that you can say that a given matrix is similar to one which hasthe diagonal entries permuted in any order desired. Then use the above problem andblock multiplication to show that if the A has k nonzero entries, then it is similar toa matrix which has k — 1 nonzero entries. Finally, when A is similar to one which hasat most one nonzero entry, this one must also be zero because of the condition on thetrace.50. tAn n x n matrix X is a comutator if there are n x n matrices A, B such that X =AB — BA. Show that the trace of any comutator is 0. Next show that if a complexmatrix X has trace equal to 0, then it is in fact a comutator. Hint: Use the aboveproblem to show that it suffices to consider X having all zero entries on the maindiagonal. Then define2 Xijp spe ysA= By = i=j if *Jme Oifi=76.11 Cauchy’s Interlacing Theorem for EigenvaluesRecall that every Hermitian matrix has all real eigenvalues. The Cauchy interlacing theoremcompares the location of the eigenvalues of a Hermitian matrix with the eigenvalues of aprincipal submatrix. It is an extremely interesting theorem.Theorem 6.11.1 Let A be a Hermitian n x n matrix and lety Bwhere B is (n—1) x (n—1). Let the eigenvalues of B be py < pg < +++ < by_y. Then ifthe eigenvalues of A are Ay < Ag < +++ < An, tt follows that Ay < fy < Ae < My S-+- <Myn—1 < Xn:Proof: First note that B is Hermitian becauseare(%® Y¥ \_yg_({ 2% yy B y BIt is easiest to consider the case where strict inequality holds for the eigenvalues for B sofirst is an outline of reducing to this case.There exists U unitary, depending on B such that U* BU = D whereMy 0