Linear Algebra, Theory And Applications, 2012a

7.4. SCHUR’S THEOREM 175
Proof: Let U k be an m k × m k unitary matrix such that
U ∗ k P k U k = T k
where T k is upper triangular. Then it follows that for
⎛
⎞ ⎛
U 1 ··· 0
U1 ∗ ··· 0
⎜
U ≡ ⎝
.
. .. .
⎟
⎠ , U ∗ ⎜
= ⎝
.
. .. . 0 ··· U s 0 ··· Us
∗
⎞
⎟
⎠
and also
⎛
U1 ∗ ··· 0
⎜
⎝
.
. .. . 0 ··· Us
∗
⎞ ⎛
⎟ ⎜
⎠ ⎝
⎞ ⎛
P 1 ··· ∗
.
. .. .
⎟ ⎜
⎠ ⎝
0 ··· P s
U 1 ··· 0
⎞
.
. .. . ⎟
⎠ =
⎛
⎜
⎝
⎞
T 1 ··· ∗
.
. .. .
⎟
⎠ .
0 ··· T s
Therefore, since the determinant of an upper triangular matrix is the product of the diagonal
entries,
det (A) = ∏ det (T k )= ∏ det (P k ) .
k
k
From the above formula, the eigenvalues of A consist of the eigenvalues of the upper triangular
matrices T k , and each T k has the same eigenvalues as P k .
What if A is a real matrix and you only want to consider real unitary matrices?
Theorem 7.4.6 Let A be a real n × n matrix. Then there exists a real unitary matrix Q
and a matrix T of the form ⎛
⎞
P 1 ··· ∗
⎜
T = ⎝
. ..
. ⎟
⎠ (7.12)
0 P r
where P i equals either a real 1 × 1 matrix or P i equals a real 2 × 2 matrix having as its
eigenvalues a conjugate pair of eigenvalues of A such that Q T AQ = T. The matrix T is
called the real Schur form of the matrix A. Recall that a real unitary matrix is also called
an orthogonal matrix.
Proof: Suppose
Av 1 = λ 1 v 1 , |v 1 | =1
where λ 1 is real. Then let {v 1 , ··· , v n } be an orthonormal basis of vectors in R n . Let Q 0
be a matrix whose i th column is v i . Then Q ∗ 0AQ 0 is of the form
⎛
⎜
⎝
λ 1 ∗ ··· ∗
0
. A 1
0
where A 1 is a real n − 1 × n − 1 matrix. This is just like the proof of Theorem 7.4.4 up to
this point.
Now consider the case where λ 1 = α + iβ where β ≠0. It follows since A is real that
v 1 = z 1 + iw 1 and that v 1 = z 1 − iw 1 is an eigenvector for the eigenvalue α − iβ. Here
z 1 and w 1 are real vectors. Since v 1 and v 1 are eigenvectors corresponding to distinct
eigenvalues, they form a linearly independent set. From this it follows that {z 1 , w 1 } is an
⎞
⎟
⎠