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LINEAR ALGEBRA OR MATRIX MATHEMATICS


Matrix definitions are provided in a separate section, called Matrices. A matrix is referenced by its elements:

A=⎡⎢⎢⎢⎣ a11 a12 a13       a21 a22 a23       a31 a32 a33 ⎤⎥⎥⎥⎦

This is a 3 × 3 matrix because it is composed of three columns and three rows respectively.The first number represents the number of rows, while the second number represents the number of columns. There can be an infinite number of different arrangements when dealing with matrices, this is the reason why its important to understand the basics before continuing to more complex matrix math.


ADDITION AND SUBTRACTION
Addition and subtraction of a matrix is quite simple. As a rule of subtraction and addition, when attempting to add or subtract matrices it is very important that the numbers of columns and rows of both matrices are equal. In other words, it is impossible to add or subtract a 2 × 2 matrix with a 3 × 3 matrix. For these operations, the operation is performed on each individual element. A12 is added to B12, A22 is added to B22, and so on. Consider matrices A and B where:

A = ⎡⎢⎢⎢⎣ a11 a12     a21 a22     a31 a32 ⎤⎥⎥⎥⎦ B=⎡⎢⎢⎢⎣ b11 b12     b21 b22     b31 b32 ⎤⎥⎥⎥⎦

Adding the matrices yields

A + B = ⎡⎢⎢⎢⎣ a11 + b11 a12 + b12     a21 + b21 a22 + b22     a31 + b31 a32 + b32 ⎤⎥⎥⎥⎦

and similarly B can be subtracted from A,

A - B = ⎡⎢⎢⎢⎣ a11 − b11 a12 − b12     a21 − b21 a22 − b22     a31 − b31 a32 − b32 ⎤⎥⎥⎥⎦


MULTIPLICATION
There are two different types of matrix multiplication, scalar and non scalar. Scalar multiplication refers to the instance where a scalar is multiplying a matrix. Before continuing with this section, one needs to understand that A × B ≠ B × A. Take for example 3 × A. This simply means that every row and column within the matrix will be multiplied by the scalar in front of it as shown below:

3 × A = ⎡⎢⎢⎢⎣ 3a11 3a12 3a13       3a21 3a22 3a23       3a31 3a32 3a33 ⎤⎥⎥⎥⎦

When multiplying two matrices together it is important that the inner number of the two matrices are identical. For example, multiplying a 3x3 times a 3x200 matrix will work because the number of columns of the first matrix matches the number of rows of the second matrix. One cannot multiply a matrix with n number of columns with a matrix with m number of rows. Consider the matrices D and C where:

D=⎡⎢⎣ d11 d12     d21 d22 ⎤⎥⎦ C=⎡⎢⎣ c11 c12 c13       c21 c22 c23 ⎤⎥⎦

Thus, D  ×  C = ⎡⎢⎣ d11 × c11 + d12 × c21 d11 × c12 + d12 × c22 d11 × c13 + d12 × c23       d21 × c11 + d22 × c21 d21 × c12 + d22 × c22 d21 × c13 + d22 × c23 ⎤⎥⎦


IDENTITY MATRIX
The identity matrix In receives its name because any multiplication with it leaves the matrix unchanged. In order to understand this, consider the following general example with matrix Z.

Z = ⎡⎢⎣ a b     c d ⎤⎥⎦

Z  ×  In = ⎡⎢⎣ a b     c d ⎤⎥⎦ × ⎡⎢⎣ 1 0     0 1 ⎤⎥⎦ = ⎡⎢⎣ a × 1 + b × 0 a × 0 + b × 1     c × 1 + d × 0 c × 0 + d × 1 ⎤⎥⎦ = ⎡⎢⎣ a b     c d ⎤⎥⎦

The In must be a square, diagonal matrix with elements, 1, and its n subscript is defined by the number of rows or columns it is composed of.
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