308 CHAPTER 13. MATRICES AND THE INNER PRODUCT

Lemma 13.2.12 Let A be of the form

A =

P1 · · · ∗...

. . ....

0 · · · Ps

where Pk is an mk×mk matrix. Then

det(A) = ∏k

det(Pk) .

Proof: Let Uk be an mk×mk unitary matrix such that

U∗k PkUk = Tk

where Tk is upper triangular. Then letting U denote the block diagonal matrix, having theUi as the blocks on the diagonal,

U =

U1 · · · 0...

. . ....

0 · · · Us

 , U∗ =

U∗1 · · · 0

.... . .

...0 · · · U∗s

and

U∗1 · · · 0...

. . ....

0 · · · U∗s



P1 · · · ∗...

. . ....

0 · · · Ps



U1 · · · 0...

. . ....

0 · · · Us

=

T1 · · · ∗...

. . ....

0 · · · Ts

and so

det(A) = ∏k

det(Tk) = ∏k

det(Pk) . ■

Definition 13.2.13 An n×n matrix A is called Hermitian if A = A∗. Thus a real symmetric(A = AT ) matrix is Hermitian.

Recall that from Theorem 13.2.14, the eigenvalues of a real symmetric matrix are allreal.

Theorem 13.2.14 If A is an n× n Hermitian matrix, there exists a unitary matrix U suchthat

U∗AU = D (13.10)

where D is a real diagonal matrix. That is, D has nonzero entries only on the main diagonaland these are real. Furthermore, the columns of U are an orthonormal basis of eigenvectorsfor Cn. If A is real and symmetric, then U can be assumed to be a real orthogonal matrixand the columns of U form an orthonormal basis for Rn.

Proof: From Schur’s theorem above, there exists U unitary (real and orthogonal if A isreal) such that

U∗AU = T