13.1. SCHUR’S THEOREM, HERMITIAN MATRICES 331

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

The following is the major result about Hermitian matrices. It says that any Hermitianmatrix is similar to a diagonal matrix. We say it is unitarily similar because the matrix Uin the following theorem which gives the similarity transformation is a unitary matrix.

Theorem 13.1.6 If A is an n× n Hermitian matrix, there exists a unitary matrix U suchthat

U∗AU = D (13.2)

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

where T is an upper triangular matrix. Then from Lemma 13.1.2

T ∗ = (U∗AU)∗ =U∗A∗U =U∗AU = T.

Thus T = T ∗ and T is upper triangular. This can only happen if T is really a diagonalmatrix having real entries on the main diagonal. (If i ̸= j, one of Ti j or Tji equals zero. ButTi j = Tji and so they are both zero. Also Tii = Tii.)

Finally, letU =

(u1 u2 · · · un

)where the ui denote the columns of U and

D =

λ 1 0

. . .

0 λ n

