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# section47 - Chapter 4 The Vector Space Rn MAT188H1F Lec03...

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Chapter 4: The Vector Space R n MAT188H1F Lec03 Burbulla Chapter 4 Lecture Notes Fall 2007 Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 4: The Vector Space R n Chapter 4: The Vector Space R n 4.7: Orthogonal Diagonalization Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla

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Chapter 4: The Vector Space R n 4.7: Orthogonal Diagonalization Digonalization Revisited In terms of the concepts of Chapter 4, we can now say that an n × n matrix A is diagonalizable if and only if R n has a basis consisting of eigenvectors of A . Why? A is diagonalizable if and only if there is an invertible matrix P and a diagonal matrix D such that D = P - 1 AP . The columns of P are eigenvectors of A , and P is invertible, so the eigenvectors form a basis of R n . We can also say that eigenvectors corresponding to distinct eigenvalues of A must be independent. Why? Suppose λ 1 = λ 2 and that AX 1 = λ 1 X 1 , AX 2 = λ 2 X 2 , X i = O . Suppose a 1 X 1 + a 2 X 2 = O . Then λ 1 ( a 1 X 1 + a 2 X 2 ) = O A ( a 1 X 1 + a 2 X 2 ) = O λ 1 a 1 X 1 + λ 1 a 2 X 2 = O a 1 λ 1 X 1 + a 2 λ 2 X 2 = O Subtract the last two equations: ( λ 1 - λ 2 ) a 2 X 2 = O a 2 = 0 . Then a 1 X 1 = O a 1 = 0 . So X 1 , X 2 are linearly independent. Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 4: The Vector Space R n 4.7: Orthogonal Diagonalization Diagonalization and Multiplicity The multiplicity of an eigenvalue λ is the number of times the factor x - λ divides the characteristic polynomial C A ( x ) of A . For example, if C A ( x ) = x 2 ( x - 1) 3 , then λ 1 = 0 has multiplicity 2 , and λ 2 = 1 has multiplicity 3 . In terms of multiplicity we can completely characterize when a matrix A is diagonalizable: A is diagonalizable if and only for each eigenvalue λ of A , dim E λ ( A ) = m , where m is the multiplicity of λ. For example, A = 1 2 0 1 is not diagonalizable, since dim E 1 ( A ) = dim null 0 1 0 0 = 1 = 2 . Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla
Chapter 4: The Vector Space R n 4.7: Orthogonal Diagonalization Eigenvectors of Symmetric Matrices Theorem: Let X 1 and X 2 be eigenvectors of the symmetric n × n matrix A , corresponding to the distinct eigenvalues λ 1 and λ 2 , respectively. Then X 1 and X 2 are orthogonal.

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