Linear Algebra BAS113 - Matrix Operations 1

Study Notes

Topics covered include matrices, summation on matrices, transpose of matrices, product on matrices, trace of matrices, and determinants.

Example matrices are shown:

[ 1  2  1]
[ 3  0 -√2]
[-1  4  π]

[  π  e]
[1/2  0]
[  0  0]

[1]  [4]
[3]

If a_ij where 1 ≤ i ≤ m and 1 ≤ j ≤ n is the elements of the matrix A, then A is written as:
```
A = [ a11 a12 ... a1n ]
   [ a21 a22 ... a2n ]
        ...
   [ am1 am2 ... amn ]
```
A is a matrix of size m × n, where m is the number of rows and n is the number of columns.
Capital letters (A, B, C…) are used to represent matrices.
A square matrix has the same number of rows and columns (n = m).
A 1×n matrix is a row vector.
An n×1 matrix is a column vector.

Example:

[ 3  0 ] = [ 4 -4 ] 
[-1  2 ]  [1  2 ]

are not equal because they have different dimensions
```
 a  b  ] = [ 4 -1]
[c  d] = [ 0 2],
```
are equal if a = 4, b = -1 , c = 0, d = 2

Matrix addition is defined as the addition of corresponding elements.
If matrix A = [a_ij] and matrix B = [b_ij] are of size m × n, then A + B is also of size m × n. A + B = [ a_ij + b_ij].
A matrix containing only zeros is called a zero matrix, denoted by 0.
k A is a matrix where each element is multiplied by a scalar k. ( kA = [ k a_ij] )
Matrix subtraction can be expressed as A - B = A + (-B).

Summarizes properties of matrix addition (commutative, associative, identity, and additive inverse).

If A = [a_ij] is an m x n matrix, then the transpose of A (A^T) is an n x m matrix obtained by interchanging the rows and columns of A.
(A^T)^T = A
(kA)^T = kA^T
(A + B)^T = A^T + B^T

Definition : The product of two matrices is only defined if the number of columns in the first matrix equals the number of rows in the second matrix. The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.
Note : Matrix multiplication is not commutative.

The trace of A (tr(A)) is the sum of the diagonal elements of a square matrix A.
Examples
Theorems on trace properties

For an n x n matrix A, the determinant (det A or |A|) is a scalar value.
Determinant is defined inductively
Examples are presented in detail
Note: Calculates determinants of 1x1 and 2x2 matrices, then more general approach for calculation of n x n matrix