Mathematical Background - Learning to Rank for Information Retrieval - page 258

21.2.2.10 Determinant

n × n n × n n ,

The determinant of a square matrix A ∈ R is a function det : R → R

denoted by det A or | A | , i.e.,

n

1 ) i + j a ij | det A = ( − A \ i, \ j | ( ∀ j = 1 , 2 ,...,n)

i = 1

or

n

1 ) i + j a ij | A \ i, \ j | det A = ( − ( ∀ i = 1 , 2 ,...,n).

j = 1

1 ) is the matrix obtained by deleting the i th row and the
j th column from A . Note that the above definition is recursive and thus we need to
define

(n − 1 ) × (n −

Here A \ i, \ j ∈ R

×

1 1 .

| A |= a 11 for A ∈ R

n × n , we have the following properties for the determinant:

Given A, B ∈ R

•| A |=| A T

|

•| AB |=| A || B |

•| |=

A 0 if and only if A is singular

| − 1

1
| A |

•| A = if A is nonsingular

21.2.2.11 Quadratic Form

n n × n , we call the following scalar

Given a vector x ∈ R and a square matrix A ∈ R

a quadratic form :

n n

x T Ax = a ij x i x j .

i = 1 j = 1

As we have

= x T A T x = x T 1

2 A T x,

x T Ax = x T Ax T

1

2 A +

we may as well regard the matrix in a quadratic form to be symmetric.
Given a vector x ∈ R

n and a symmetric matrix A ∈ R n × n ,wehavethefollowing

definitions:

If x T Ax > 0, • ∀ x , A is positive definite , denoted by A 0.

If x T Ax • ≥ 0, ∀ x , A is positive semidefinite , denoted by A 0.

If x T Ax < 0, • ∀ x , A is negative definite , denoted by A ≺ 0.

If x T Ax • ≤ 0, ∀ x , A is negative semidefinite , denoted by A 0.

• Otherwise, A is indefinite .