Try the Free Math Solver or Scroll down to Tutorials!

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


MATRIX ALGEBRA


Example:

From the first expression,

From the second expression

Usual practise to determine an eigenvector with length one. In the example:

Definition: Quadratic form Q(x) in k variables , where ,
is defined as

Q(x) = x'Ax where A is a fix (k × k) matrix

The quadratic form is a quadratic function of  .

Example:

Theorem Let A be a (k × k) symmetric matrix, (i. e. A = A') then A has k pairs of
eigenvalues and eigenvectors

The eigenvectors can be chosen to satisfy

and be mutually perpendicular. The eigenvectors are unique unless two or more eigenvalues
are equal.

Definition: Positive Definite Matrices Let A be a symmetric matrix (k × k). A is
said to be positive definite if
x'Ax > 0 for all x ∈ Rk x ≠ 0.

A is positive semi-definite if
x'Ax ≥ 0 for all x ∈ Rk.

Theorem: Spectral decomposition of a symmetric (k × k) matrix A is given:

are eigenvalues of A and are the associated normalized eigenvectors.

Theorem A, (k × k) symmetric matrix is positive definite if and only if every eigenvalue
of A is positive.

A is positive semi-definite if and only if every eigenvalue of A is nonnegative.

Proof: Trivial from the spectral decomposition theorem.

where . Choosing x = , j = 1, . . . , k) follows the theorem.

Another form of the spectral decomposition:

INVERZE AND SQUARE ROOT OF A POSITIVE DEFINIT MATRIX
By the spectral decomposition, if A is positive definite:



Inverse of A If A is positive definite, then the eigenvaluse of A are
i = 1, . . . , k. The inverse of A is where .

Because;

Square Root Matrix (defined for positive semi-definite A)

Notice:

MATRIX INEQUALITIES AND MAXIMIZATION

Cauchy-Schwarz Inequality:
Let b, d ∈ Rp, then

with equality if and only if b = cd.

Proof: The inequality is obvious when one of the vectors is the zero one. For b − xd ≠ 0 we
have that,

Adding and substratcting , we have that

Choosing x = b'd/d'd follows the statement.

Extended Cauchy–Schwartz Inequality:
Let b, d ∈ Rp and let B be a positive definite
(p × p) matrix. Then



with equality if and only if b = cB-1d for some constant c.

Maximization of Quadratic forms on the Unit Sphere:

Let B be a (p × p) positive definite matrix with eigenvectors and associated
eigenvalues eigenvectors, where . Then

Moreover

and the equality is attained when .