6.2 SPECIAL MATRICES
Inverse of a product: (AB)−1 = B −1 A−1 . Transpose of a product: (AB)T = B T AT .
A matrix Q is said to be orthogonal if
QT Q = QQT = I
That is, the transpose of an orthogonal matrix is its inverse, Q−1 = QT .
The columns of an orthogonal matrix are vectors that form an orthonormal set.
If A = AT is a real symmetric matrix then:
• All eigenvalues of A are real.
• There is always a full set of eigenvectors.
• Eigenvectors corresponding to different eigenvalues are orthogonal.
• If Q is the matrix of unit eigenvectors of A then Q is an orthogonal matrix and the
diagonalisation of A is
QT AQ = D
.
In this lecture we will look at some important properties of invertible matrices, orthogonal
matrices, and symmetric matrices.
Example 1 Prove that for any square invertible matrices A and B of the same dimension
(AB)−1 = B −1 A−1
1
Example 2 If A is an invertible matrix with eigenvector v and corresponding eigenvalue
λ prove that v is also an eigenvector of A−1 and find the corresponding eigenvalue.
1
λ
Example 3 Suppose that A is a matrix with eigenvector v and corresponding eigenvalue
λ and that P −1 AP = D. Prove that P −1 v is an eigenvector of D also with eigenvalue λ.
2
An orthonormal set is a set of vectors that are (i) perpendicular to each other and (ii)
have unit length. For example, if u, v and w are an orthonormal set then
u · v = uT v = 0,
v · w = vT w = 0,
u · w = uT w = 0.
u · u = uT u = 1,
v · v = vT v = 1,
w · w = wT w = 1.
Note that we have used the fact that dot products can always be expressed as matrix
products. For example
2
4
1 · 0 = 8 + 0 + 15 = 23
5
3
T
2
4
1 0 =
5
3
Example 4 Show that
1
√
6
1
√
6
2
√
6
,
−2
√
5
0
1
√
5
2 1 5
4
0 = 23
3
1
√
30
−5
,
√
form an orthonormal set.
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