# Mechengg

## Mechengg

• Submitted By: Ggggggj
• Date Submitted: 07/05/2016 8:39 AM
• Words: 748
• Page: 3

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.
 

 ...