Matrix Algebra Definitions Operations Matrix algebra is a means of making calculations upon arrays of numbers (or data). Most data sets are matrix-type What is it? Why use it? Matrix algebra makes mathematical expression and computation easier..

Definitions - Matrix A matrix is an array of numbers A = Denoted with a bold Capital letter All matrices have an order (or dimension): that is, the number of rows  the number of columns. So, A is 2 by 3 or (2  3).

Definitions A square matrix is a matrix that has the same number of rows and columns (n  n). Two matrices are equal if and only if they both have the same number of rows and the same number of columns their corresponding elements are equal. or two matrices A and B of same order m × n are said to be equal if and only if all of their components are equal: a ij = b ij, for all i = 1,...m, j = 1,... n. We then write A = B. Matrix Equality

Matrix Operations Addition and Subtraction Multiplication Transposition

Addition and Subtraction Two matrices may be added (or subtracted) if they are the same order. Simply add (or subtract) the corresponding elements. So, A + B = C yields Where

Addition Subtraction

Matrix Multiplication To multiply a scalar times a matrix, simply multiply each element of the matrix by the scalar quantity To multiply a matrix times a matrix, we write AB (A times B) This is pre-multiplying B by A.

Matrix Multiplication Thus where

Matrix Multiplication- an example Thus where

Properties AB does not necessarily equal BA (BA may even be an impossible operation) For example, A  B = C (2  3)  (3  2) = (2  2) B  A = D (3  2)  (2  3) = (3  3)

Computation: A x B = C [3 x 2][2 x 3] A and B can be multiplied [3 x 3]

If A is an m × n matrix, the transpose of A, written, is the n × m matrix whose rows are just the columns of A in the same order. In other words, the first row of is the first column of A, the second row of is the second column of A, and so on. Transpose

The Transpose of a Matrix: A' The transpose of a matrix is a new matrix that is formed by interchanging the rows and columns. The transpose of A is denoted by A' or (A T ) Example Thus, If A = A', then A is symmetric