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Unformatted text preview: Universityof Houston, Department of Mathematics Numerical Analysis, Fall 2005 Chapter 6 Numerical solution of eigenvalue problems 6.1 Theoretical foundations Definition 6.1 Eigenvalue, eigenvector, spectrum Let A ∈ C n × n . Anumber λ ∈ C is called an eigenvalue of A , if there exists a vector x ∈ C n , x negationslash = such that ( ∗ ) Ax = λ x . The vector x is called an eigenvector associated with λ . The set of all eigenvalues λ of A is said to be the spectrum of A and will be denoted by σ ( A ) . An immediate consequence of ( ∗ ) is the following: λ ∈ σ ( A ) ⇐⇒ N ( A − λ I ) := { x ∈ C n \{ } ( A − λ I ) x = } negationslash = ∅ ⇐⇒ det ( A − λ I ) = . Definition 6.2 Characteristic polynomial, multiplicities The polynomial ϕ A ∈ P n ( C ) given by ϕ A ( λ ) = det ( A − λ I ) , λ ∈ C is called the characteristic polynomial of A . Universityof Houston, Department of Mathematics Numerical Analysis, Fall 2005 If σ ( A ) = { λ 1 ,... λ k } , λ i negationslash = λ j , 1 ≤ i negationslash = j ≤ k , then ϕ A can be written as ϕ A ( λ ) = ( − 1 ) n ( λ − λ 1 ) m 1 ( λ − λ 2 ) m 2 ... ( λ − λ k ) m k , m i ∈ lN , 1 ≤ i ≤ k , k summationdisplay i = 1 m i = n . The number m i is called the algebraic multiplicity of the eigenvalue λ i and will be denoted by σ ( λ i ) := m i . On the other hand, ρ ( λ i ) := dim ( N ( A − λ i I ) is said to be the geometric multiplicity of λ i . There holds 1 ≤ ρ ( λ i ) ≤ σ ( λ i ) ≤ n . Definition 6.3 Similarity transformation Assume that T ∈ C n × n is regular. Then, the mapping A mapsto−→ B := T − 1 AT is called a similarity transformation . The matrices A and B are said to be similar . Lemma 6.4 Invariance properties of similarity tranbsformations For A ∈ C n × n , the spectrum σ ( A ) , the characteristic polynomial ϕ A and the multiplicities ρ ( λ ) , σ ( λ ) , λ ∈ σ ( A ) , are invariant with respect to similarity transformations . Universityof Houston, Department of Mathematics Numerical Analysis, Fall 2005 Similarity transformations are used to transformmatrices to a simpler form, called canonical forms . Froma numerical point of view, we are particularly interested in unitary transforma tions U (i.e., U H = U − 1 ) whose spectral condition number κ 2 ( U ) is equal to 1 . Theorem6.5 The Schur normal form For each matrix A ∈ C n × n with eigenvalues λ i ∈ C , 1 ≤ i ≤ n , there exists a unitary matrix U ∈ C n × n such that U H AU = λ 1 ⋆ · · · ⋆ λ 2 · · · · ⋆ 0 0 λ n ....
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This note was uploaded on 02/21/2012 for the course ENG 3000 taught by Professor Staff during the Spring '12 term at University of Houston.
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