Properties of Matrix Operations 2010, 14, Sep. Ki-Seung Lee

Properties of Matrix Addition A+B = B+A A+(B+C)=(A+B)+C (cd)A = c(dA) 1A=A c(A+B) = cA+cB (c+d)A = cA +dA A+0 mn =A A+(-A) = 0 mn If cA=0 mn then c=0 or A=0 mn. Commutative Associative Scalar Associative Scalar identity Scalar distributive 1 Scalar distributive 2 Additive identity Additive Inverse Scalar cancellation property

Properties of Matrix Multiplication A(BC) = (AB)C A(B+C) = AB +AC (A+B)C = AC+BC c(AB) = (cA)B=A(cB) AI n = A I m A = A assuming A is m by n and a ll operations are defined. –Associative –Left distributive –Right Distributive –Scalar Associative –Multiplicative Identity

Using Properties to Prove Theorems Using these properties we can prove the following the orem (which we have already been assuming). Theorem: For a system of linear equations in n variables, precisely one of the following is tr ue: 1. The system has exactly one solution. 2. The system has an infinite number of solutions. 3. The system has no solutions.

The Transpose of a Matrix We will find it useful at times to talk about the transpose of a matrix. Given an m by n matrix A, we define A T ( A tran spose ) to be the n by m matrix:

Properties of Transposes 1. (A T ) T = A 2. (A + B) T = A T +B T 3. (cA) T = c(A T ) 4. (AB) T = B T A T Transpose of a transpose Transpose of a sum Transpose of a scalar prod uct Transpose of a product

What about Mult. Inverses For an n by n matrix A, can we find an n by n matrix A -1 so that AA -1 =A -1 A=I n ? Does this always work?

Properties of Inverse Matrices

Definition Last time we said the the inverse of an n by n matrix A is an n by n matrix B where, AB = BA = I n. We also talked about how to find the inverse o f a matrix and said that not all matrices have in verses (some are singular) so won’t review that here.

Properties of Inverses 1. If A is an invertible matrix then its inverse is u nique. 2. (A -1 ) -1 = A. 3. (A k ) -1 = (A -1 ) k ( we will denote this as A -k ) 4. (cA) -1 = (1/c)A -1, c ≠ ( A T ) -1 = (A -1 ) T.

Some theorems involving Inverses 1. If A and B are invertible matrices then, (AB) -1 = B -1 A If C is an invertible matrix then the following prop erties hold. a) If AC = BC then A = B. b) If CA = CB then A = B. 3. If A is an invertible matrix, then the system of equ ations Ax = b has a unique solution given by x = A -1 b.

Elementary Matrices An n by n matrix is called an elementary matr ix if it can be obtained from I n by a single elem entary row operation. These matrices allow us to do row operations with matrix multiplication.

Representing Elementary Row Operations Theorem: Let E be the elementary matrix obtain ed by performing an elementary row operation on I n. If that same row operation is performed on an m by n matrix A, then the resulting matri x is given by the product EA.

Row equivalent matrices Let A and B be m by n matrices. Matrix B is ro w equivalent to A if there exists a finite numbe r of elementary matrices E 1, E 2,... E k such that B = E k E k-1... E 2 E 1 A.

LU Factorizations

Review of Last Time An elementary matrix is a matrix that can be o btained from an identity matrix by applying a s ingle row operation The inverse of an elementary matrix is also an elementary matrix Doing row operations can be seen as multiplyi ng by an elementary matrix

Fact from last time A square matrix A is row equivalent to the iden tity matrix if and only if it can be written as a p roduct of elementary matrices. Theorem: A square matrix A is invertible if an d only if it can be written as the product if ele mentary matrices.

The BIG Theorem The following statements are equivalent (TFS AE) for any n by n square matrix A. 1. A is invertible. 2. Ax = b has a unique solution for any n by 1 matrix b. 3. Ax = 0 has only the trivial solution. 4. A is row-equivalent to I n. 5. A can be written as the product of elementary matr ices.

A few obvious definitions A matrix A is said to be upper triangular if a k,,l  0 implies l ≥ k. A matrix A is said to be lower triangular if a k,,l  0 implies k ≥ l.

Why do we care??? If we could somehow factor an n by n matrix A into a lower triangular matrix L and an upper tr iangular matrix U, A = LU then we can solve any system of equations Ax = b without doing row operations.

So how can we find LU factorizations?? Let’s start by trying to row reduce A to an uppe r triangular matrix. What could happen to make this not work?