**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 ∈ R^{k} x ≠ 0.

A is positive semi-definite if

x'Ax ≥ 0 for all x ∈ R^{k}.

**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:

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 ∈ R

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 ∈ R

(p × p) matrix. Then

with equality if and only if b = cB

Let B be a (p × p) positive definite matrix with eigenvectors and associated

eigenvalues eigenvectors, where . Then

Moreover

and the equality is attained when .