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Matrix algebra summary

Nikolai Shokhirev

Introduction

Object definitions

A matrix is a  rectangular array of numbers () i, jx i, j , i = 1, .., Nj = 1, .., M :

     (1)

Here N is the number of rows and M is the number of columns ( N by M matrix).

The transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. It is usually denoted by the superscript T

(X T ) i, j = (X) j, i       (2)

Obviously (X T ) is an M by N matrix.

A vector is a linear array of numbers y i, j , i = 1, .., N:

     (3)

It can be considered as an N by 1 matrix. It is also called as a column vector.

Another particular case of a matrix is a row vector:

     (4)

It can be considered as an 1 by M matrix and it is the transpose of a column vector.

Operations

For any object (matrix or vector) a multiplication by a scalar is defined as a multiplication of each element (component) by the scalar:

(c · X ) i, jc · x i, j        (5)

For the objects of the same dimension the addition and subtraction are defined as 

 (A ± B) i, j = a i, j ± b i, j      (6)

The product C of two matrices A and B is defined as

       (7)

 Eq. (7) implies the following relationship between the dimensions of the matrices

 Matrix   Dimensions 
A N by K
B K by M
C N by M

Matrix multiplication is associative:

(A · B) · C = A · (B · C) = A · B · C      (8)

However, matrix multiplication in general is not commutative:

A · BB · A      (9)

 In the case M = 1 and N =  1 Eq. (7) reduces to the dot product of inner product of two vectors. In this case C is an 1 by 1 matrix, i.e. a scalar:

       (10)

 

 In the case K = 1 Eq. (7) reduces to the direct or outer product of two vectors. In this case C id an N by M matrix:

       (11)

 Particular cases

 The matrix X (1) is square if N = M

A diagonal matrix is a square matrix A of the form

a i, j = a i  δ i, j       (12)

where δ i, j  is the Kronecker delta

       (13)

 

Useful formulae

  Transposition of a product

 (A · B) T = B T · A T       (14)

  Inversion of a product

       (15)

Woodbury matrix identity

       (16)

 

      

References

  1. Woodbury matrix identity http://en.wikipedia.org/wiki/Woodbury_matrix_identity
  2. Matrix Inverse in Block Form by J.-S. Roger Jang, 2001-03-21 http://www.cs.nthu.edu.tw/~jang/book/addenda/matinv/matinv/

 

© Nikolai Shokhirev, 2003 - 2017

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