MAT223H5 Study Guide - Final Guide: Diagonalizable Matrix, If And Only If, Determiner

• School: UTM
• Department: Mathematics
• Course: MAT223H5
• Professor: Lukic

Document Summary

Let a and b be similar matrices, that is, a=p-1bp, then a and b share the same eigenvalues with the same multiplicities. An nxn matrix is diagonalizable when iff the dimension of the basis of the eigenspace associates with each eigenvalue is equal to the multiplicity of that eigenvalue. dim(e (a))=mult( ) Extension: if a is an nxn matrix and has n distinct eigenvalues, a is diagonalizable. A is invertible iff zero is not an eigenvalue of a. Ak=udku-1 where u is invertible and d is diagonal. For any complex number ( ) z=a+bi, define|a+bi|= For any complex number ( ) z=a+bi, it has a polar form z=r e of z i where r=|a+bi|, and is called the argument. Any polynomial of degree one or more with a complex coefficient has a complex root. e. g. x2-2ix-5=0; x=i+2. Given a polynomial p(x), if x=z (z is a complex number) is a root, than the conjugate of z (