# Linear algebra

by Nikolai V. Shokhirev

- Linear algebra

### Object definitions

A matrix is a  rectangular array of numbers (X T ) 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, j =  c · 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)

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:

 (9)

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:

 (10)

### 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 (11)

where δ i, j  is the Kronecker delta

 (12)

### Useful formulae

 (A · B) T = B T · A T (13)

In progress . . .

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- Linear algebra