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.

...