by Nikolai V. Shokhirev
Up: ABC Tutorials
 Linear algebra

A matrix is a rectangular array of numbers (X^{ T }) _{ i}_{, j} = x_{ i}_{, j} , i = 1, .., N; j = 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.
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) 
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) 
(A · B)^{ T} = B^{ T} · A^{ T}  (13) 
In progress . . .
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Up: ABC Tutorials
 Linear algebra

©Nikolai V. Shokhirev, 20052008