Page 288 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Proof: The rows , …, of B form a basis for C . Let , …, be the vectors obtained from ,
n
1 n 1 n 1
…, by the Gram-Schmidt process. Then, for 1 k n, { , …, } is an orthogonal basis for the
n 1 k
subspace spanned by { , …, }, and
1 k
( | )
= k j j .
k k 2
j k j
Hence, for each k there exist unique scalars C such that
k j
= C j .
k k k j
j k
Let U be the unitary matrix with rows

, ,
1 n
1 n
and M the matrix defined by

1
C k , j if j k
k
1
M = , if j k
k j
k
0, if j k

Then M is lower-triangular, in the sense that its entries above the main diagonal are 0. The
entries M of M on the main diagonal are all > 0, and
kk
n
k M , 1 k n .
= kj j
k j 1
Now these equations simply say that
U = MB.
+
To prove the uniqueness of M, let T (n) denote the set of all complex n × n lower-triangular
+
matrices with positive on the main diagonal. Suppose M and M are elements of T (n) such that
1 2
M B is in U(n) for i = 1, 2. Then because U(n) is a group
i
(M B) (M B) –1 = M M –1
1 2 1 2
lies in U(n). On the other hand, although it is not entirely obvious, T (n) is also a group under
+
matrix multiplication. One way to see this is to consider the geometric properties of the linear
transformations
+
X MX, (M in T (n))
+
–1 –1
–1
on the space of column matrices. Thus M , M M , and (M M ) are all in T (n). But, since
–1
2 1 2 1 2
–1
–1 –1
M M is in U(n), (M M ) = (M M )*. The transpose or conjugate transpose of any lower-
–1
1 2 1 2 1 2
triangular matrix is an upper-triangular matrix. Therefore, M M is simultaneously upper and
–1
1 2
lower-triangular, i.e., diagonal. A diagonal matrix is unitary if and only if each of its entries on
the main diagonal has absolute value 1; if the diagonal entries are all positive, they must
–1
equal 1. Hence M M = I and M = M .
1 2 1 2
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