What is the matrix calculation formula?

What is the matrix calculation formula?

The most basic operations of the matrix include matrix addition (subtraction), multiplication and transposition. There are more than one type of operations called "matrix addition," "multiplication," and "transpose." [4]

Example: Given m × n matrices A and B, you can define their sum A + B as an m × n matrix, etc. i, j terms are (A + B)[i, j] = A[i, j] + B[i, j].

Alternative additions can be found in matrix additions.

If a matrix A and a number c are given, a scalar product cA can be defined, where (cA)[i, j] = cA[i, j]. E.g

These two operations make M(m, n, R) a real linear space, and the dimension is mn.

If the number of columns in one matrix is ​​equal to the number of rows in another matrix, the product of the two matrices can be defined. For example, A is an m×n matrix and B is an n×p matrix. They are products. AB is an m×p matrix, where

(AB)[i, j] = A[i, 1] * B[1, j] + A[i, 2] * B[2, j] + ... + A[i, n] * B[ n, j] for all i and j.

E.g

This multiplication has the following properties:

(AB)C = A(BC) For all k × m matrices A, m × n matrices B and n × p matrices C ("joining law").

(A + B) C = AC + BC For all m × n matrices A and B and n × k matrices C ("distribution law").

C(A + B) = CA + CB For all m × n matrices A and B and k × m matrices C ("distribution law").

It should be noted that the replaceability is not necessarily true, that is, there are matrices A and B such that AB ≠ BA.

For other special multiplications, see Matrix Multiplication.