Matrices Square is Good! Copyright © 2014 Curt Hill

Introduction Matrices seem to have been developed by Gauss, for the purpose of solving systems of simulteneous linear equations –Before 1800s they are known as arrays –A Chinese text discusses their use for linear equations that dates to BC times We will also use but not for that purpose, but lets start with that Copyright © 2014 Curt Hill

A matrix When we have systems of linear equations the first step is to put them in standard form: a 1 x+b 1 y=c 1 a 2 x+b 2 y=c 2 In standard form we always write the equations in the same way On the left hand side are all the variable and on the right only constants The variables are in the same order Copyright © 2014 Curt Hill

Consider Two equations 3x - 7y = 1 2x - 3y = -1 This is now in standard form: All variables to left of = All constants to right Xs always come first and Ys next The first column is the X The second column is the Y The third the constant The next step is to render this in terms of a matrix Copyright © 2014 Curt Hill

Matrix Example Copyright © 2014 Curt Hill

Gaussian Elimination Copyright © 2014 Curt Hill

In General We may have any number of rows and columns A column matrix has one column but multiple rows –Also known as a column vector A row matrix has one row but multiple columns –Also known as a row vector A square matrix has the same number of rows and columns Copyright © 2014 Curt Hill

Elements Copyright © 2014 Curt Hill

Arithmetic We also have the notion of matrix arithmetic –At least addition, subtraction, product (multiplication) In the rationals and real numbers any two items may participate in any operation This is not true for matrix arithmetic –Two matrices must be compatible –The rules of compatibility vary by operation Copyright © 2014 Curt Hill

Matrix Addition Two matrices are compatible for addition if they have the same sizes –The two must have the same number of columns and same number of rows The sum of two matrices is a new matrix with the same size of the two that were added Each position has the sum of the corresponding positions of the two Copyright © 2014 Curt Hill

Addition Again Copyright © 2014 Curt Hill

Audience Participation Copyright © 2014 Curt Hill

Multiplication Multiplication is somewhat more complicated There are two things to consider: –Matrix compatibility –How the operation proceeds Neither is as easy as addition Matrix multiplication is not commutative We denote the product of matrix A and matrix B as AB Copyright © 2014 Curt Hill

Compatibility The matrix AB may only be computed if the number of columns of A is identical with the number of rows of B –The number of rows of A and number of columns of B do not matter Thus if A is m  k and B is k  n the two are compatible Square matrices of the same size are compatible, but all other compatible matrices have a different shape Copyright © 2014 Curt Hill

Process If A is m  k and B is k  n the result, AB is m  n The new matrix has each element as a sum of products A column from the left times the row from right C=AB C ij = A i1 B 1j + A i2 B 2j + … + A ik B kj Copyright © 2014 Curt Hill

Example Copyright © 2014 Curt Hill

Exercise Copyright © 2014 Curt Hill

Now Then Copyright © 2014 Curt Hill

Something else Copyright © 2014 Curt Hill

Identity Matrix A square matrix with ones on the diagonal and zeros else where is an identity matrix When multiplied with a compatible matrix it gives the matrix back –Similar to one with respect to multiplication and zero with respect to addition –One is the unit of multiplication as is zero to addition Copyright © 2014 Curt Hill

Online Resources The following links may be helpful: First two on a Google Search Copyright © 2014 Curt Hill

Transpose Copyright © 2014 Curt Hill

Inverse Matrix Two square matrices of same size are inverses of one another if their product is the identity matrix –They are invertable Thus AB = BA = I An inverse is unique We also use the notation A -1 for the inverse of A In linear algebra a coefficient matrix that is invertable has a unique solution Copyright © 2014 Curt Hill

Zero-One Matrix Any matrix, such as the identity matrix, that has only zeros and ones Often matrices representing Boolean values The join of two identically sized zero one matrices is the corresponding elements ORed The meetjoin of two identically sized zero one matrices is the corresponding elements ANDed Copyright © 2014 Curt Hill

Example Copyright © 2014 Curt Hill

Boolean Product Similar to matrix multiplication except the adds are replaced by ORs and multiplies by ANDs The symbol is  Copyright © 2014 Curt Hill

Example Copyright © 2014 Curt Hill

Powers Raising a number to a power is merely repeated multiplication We may also raise square zero-one matrices to a power Conjunction and disjunction are associative so Boolean product is as well We define the zeroth power to be an identity matrix Copyright © 2014 Curt Hill

Exercises 3, 7, 11, 15, 27, 29 Copyright © 2014 Curt Hill