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Lect28

Lect28 - Chapter 6 Eigenvalues Eigenvectors Diagonalization...

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Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenspaces Theorem A Example . Suppose λ 6 = μ are two eigenvalues of A . Write W λ and W μ for the corresponding eigenspaces. Show that W λ W μ = { 0 } .

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Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenspaces Theorem A Example . Suppose λ 6 = μ are two eigenvalues of A . Write W λ and W μ for the corresponding eigenspaces. Show that W λ W μ = { 0 } .
Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenspaces Theorem A Example . Suppose λ 6 = μ are two eigenvalues of A . Write W λ and W μ for the corresponding eigenspaces. Show that W λ W μ = { 0 } . By definition dim W λ 1 . How big can dim W λ be? A trivial bound is dim W λ n where n is the order of A .

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Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenspaces Theorem A Example . Suppose λ 6 = μ are two eigenvalues of A . Write W λ and W μ for the corresponding eigenspaces. Show that W λ W μ = { 0 } . By definition dim W λ 1 . How big can dim W λ be? A trivial bound is dim W λ n where n is the order of A . Let the characteristic polynomial of A be p ( x ) . Suppose λ is an eigenvalue of A with multiplicity d λ , i.e. p ( x ) = ( x - λ ) d λ q ( x ) where q ( x ) is a polynomial and q ( λ ) 6 = 0 .
Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenspaces Theorem A Example . Suppose λ 6 = μ are two eigenvalues of A . Write W λ and W μ for the corresponding eigenspaces. Show that W λ W μ = { 0 } . By definition dim W λ 1 . How big can dim W λ be? A trivial bound is dim W λ n where n is the order of A . Let the characteristic polynomial of A be p ( x ) . Suppose λ is an eigenvalue of A with multiplicity d λ , i.e. p ( x ) = ( x - λ ) d λ q ( x ) where q ( x ) is a polynomial and q ( λ ) 6 = 0 . Theorem A dim W λ d λ

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Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenspaces Algebraic Multiplicity & Geometric Multiplicity Definition We call d λ the algebraic multiplicity of λ . We call dim W λ the geometric multiplicity of λ .
Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenspaces Algebraic Multiplicity & Geometric Multiplicity Definition We call d λ the algebraic multiplicity of λ .

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