by Nikolai V. Shokhirev
Up: ABC TutorialsA matrix is a rectangular array of numbers (X T ) i, j = x i, j , i = 1, .., N; j = 1, .., M :
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(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:
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(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:
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(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
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(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:
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(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:
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(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
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(12) |
| (A · B) T = B T · A T | (13) |
In progress . . .
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- Linear algebra
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©Nikolai V. Shokhirev, 2005