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ENGG 1500 (23)
Lecture 15

# Lecture 15 -Diagonalization - March 7 2013.pdf

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School
University of Guelph
Department
Engineering
Course
ENGG 1500
Professor
Medhat Moussa
Semester
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 = det⁡𝐴 𝑐𝑜𝑓⁡𝐴 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 𝑡𝑟⁡𝐴 = 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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