Page 322 - A Course in Linear Algebra with Applications

306 Chapter Nine: Advanced Topics

Let UQ denote the matrix (X\... X n); then UQ is unitary
since its columns form an orthonormal set. Now

= U* 0{ ClX x) = ci(C/ 0*Xi).
U* QAX X

Also X*X l=0iii>l, while X?X X = 1. Hence

/ c i \
0
U^AX X = Cl
X X
. n l J W

Since

UZAU Q = U* QA{X 1 ...X n) = (U£AX 1 U* 0AX 2 ... U£AX n),

we deduce that

ci B
U^AUo =
0 Ai

where A\ is a matrix with n — 1 rows and columns and 5 is
an (n — l)-row vector.
We now have the opportunity to apply the induction
hypothesis on n; there is a unitary matrix U\ such that
C/*i4it/i = Ti is upper triangular. Put

1 0
C/ 2 =
0 C/i

which is surely a unitary matrix. Then let U — VQU^'I this
also unitary since U*U = U^(U^U 0)U 2 = U;U 2 = I. Finally

^M£/ = C^(^ 0M^o)y'2 = ^ ( C 0 1 f j ^ ,