1/15 Fun Matrix Facts and Tricks Ben G. Fitzpatrick Department of Mathematics Loyola Marymount University One LMU Drive Los Angeles, CA 90045

2/15 Matrix equations System of equations Matrix form Solve by inversion

3/15 Matlab matrix equations A=[1,1 ; 2,3]; –Comma continues row, semicolon goes to next row b=[10;13]; –Again, semicolon goes to next row, making b a column vector x=inv(A)*b; –Computes the inverse of A and multiplies it to b. x=A\b; –Also works, sort of like “dividing by A” without computing “one over A”

4/15 Matrix equations, cont’d System of equations Matrix form Solve by inversion –No solution –TRY IT IN MATLAB??!?!?!

5/15 Matrix equations, theory A square matrix has an inverse If and only if All the columns are linearly independent If and only if The only solution of If and only if

6/15 Determinants If Like Wronskian!!!

7/15 Determinants Let Define Then

8/15 Determinant example

9/15 Why are we doing this? To help with In fact, you can do the taylor series:

10/15 Eigenvalues If there is a number and a non-zero vector so that Then is called an eigenvalue of A and is called an eigenvector Note that

11/15 Eigenvalues

12/15 Eigenvectors

13/15 Eigenvectors

14/15 A factorization of matrices

16/15 Let’s Use MATLAB Try your matrix… look at the frequencies of vibration! A=[-9,-11;-11,-9] [X,L] = eig(A) Xi = inv(X) Lcheck=Xi*A*X

17/15 Two methods for exponential of matrix Taylor’ series: Eigenvalues and eigenvectors

18/15 From the book Section 3.1: matrix-vector basics –Know how to add, multiply matrices –Know how to set up matrix versions of systems –Know how to convert higher-order differential equations to systems of first order ones. Section 3.8: eigenvectors and eigenvalues –What they are –Use determinants to find them –Connection to matrix exponential