A matrix is a rectangular array of numbers () _{ 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) |
However, matrix multiplication in general is not commutative:
A · B ≠ B · 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) |
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) |
Transposition of a product
(A · B)^{ T} = B^{ T} · A^{ T} | (14) |
Inversion of a product
(15) |
Woodbury matrix identity
(16) |
© Nikolai Shokhirev, 2003 - 2017