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Lecture 15

# Lecture 15 -Diagonalization - March 7 2013.pdf

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University of Guelph

Engineering

ENGG 1500

Medhat Moussa

Winter

Description

Lecture 15
Diagonalization LetAbe an n x n matrix having a basis {v , …, 1
v } of eigenvectors ofA. Let the
n
corresponding eigenvalues be denoted , …, 1
, respectively.
n
Let P = [v 1 v ] n
AP =A[v … v ] = PD => P AP = D -1
1 n
2 Diagonlizable (page 300 in text):
If there exists an invertible matrix P and diagonal
-1
matrix D such that P AP = D, then we say thatA is
diagonalizable and that the matrix P diagonalizesA
to its diagonal form D
3 Inverting a Matrix using Cofactors
first need to find the cofactor matrix of A
(cof A) = C
ij ij
then need to calculate det A
−1 1 𝑇
𝐴 = det𝐴 𝑐𝑜𝑓𝐴
4 Example
Find the cofactor matrix and inverse of
1 2 0
𝐴 = −1 0 1
0 3 1
5 Theorem 1 (page 300 in text)
-1
IfAand B are n x n matrices such that P AP = B for
some invertible matrix P, thenAand B have
1) The same determinant
2) The same eigenvalues
3) The same rank
4) The same trace, where the 𝑛race of a matrix A is defined by
𝑡𝑟𝐴 = 𝑎 𝑖𝑖
𝑖=1
6 Similar Matrices (page 300 in text)
If A and B are n x n matrices such that P AP = B for
some invertible matrix P , thenA and B are said to be
similar
7 Theorem 2: Diagonalization Theorem (page
301 in text)
An n x n matrix A can be diagonalized if and only if
n
there exists a basis for R of eigenvectors of A. If
such a basis {v , …, v } exists, the matrix
1 n
P = [v 1 v ] dnagonalizesA to a diagonal matrix
D = diag( 1, …,n), where iI an eigenvalue of A
corresponding to v for 1 ≤ i ≤ n
i
8 Corollary 3 (page 301 in text):
A matrix A is diagonalizable if and only if every
eigenvalue of a matrix A has it

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