Introduction to Computational Linear Algebra

110 Introduction to Computational Linear Algebra

Proof. Let us consider the situation when A is Hermitian: A

∗

= A. Its Schur

factorization T = U

∗

AU satisfies: T

∗

= (U

∗

AU)

∗

= U

∗

AU = T . Conse-

quently, the matrix T is Hermitian and upper-triangular; the only possibility

of such a matrix is to be a diagonal matrix. Moreover, its diagonal entries are

equal to their conjugate and this proves that they are real.

Theorem 5.3 (Real Schur Factorization) Let A ∈ R

n×n

. It is orthogo-

nally similar to a quasi-upper triangular matrix: there exists an orthogonal

matrix Q ∈ R

n×n

such that the matrix T = Q

T

AQ is block-upper triangular,

with diagonal blocks of order 1