Summary linear algebra (notes, exercises, correction)

CHAPTER 7 SYMMETRIC MATRICES AND QUADRATIC
FORMS

7.1 Diagonalization of Symmetric Matrices


• Symmetric Matrix


• A matrix A is symmetric if AT = A.
• A matrix A is anti-symmetric if AT = −A.




 
6 −2 −1
 
• Example If possible, diagonalize the matrix A =  −2 6 −1 .
 
 
−1 −1 5




1

,2

• Theorem

If A is a symmetric matrix, then any two eigenvectors from different
eigenspaces are orthogonal.




• Theorem

An n×n matrix is orthogonally diagonalizable if and only if A is a symmetric
matrix.

, 3
 
3 −2 4
 
• Example Orthogonally diagonalize the matrix A =  −2 6 2 , whose
 
 
4 2 3
characteristic polynomial is −(λ − 7)2 (λ + 2).

, 4

• The Spectral Theorem

An n × n symmetric matrix A has the following properties:
(a) A has n real eigenvalues, counting multiplicities.
(b) The dimension of the eigenspace for each eigenvalue λ equals the multi-
plicity of λ as a root of the characteristic equation.
(c) The eigenspces are mutually orthogonal.
(d) A is orthogonally diagonalizable.




• Spectral Decomposition


If A is a symmetric matrix, then A = P DP −1 = P DP T , where P =
h i
u1 . . . un ,
  
λ1 . . . 0 uT
h i  1
A = P DP T = u1 . . . un  ... . . . ...   ... 
  
  
0 . . . λn uTn

=λ1 u1 uT1 + . . . + λn un uTn .

The last expression is called a spectral decomposition of A.