Draft. Lecture 14 Eigenvalue Problems. MATH 562 Numerical Analysis II. Songting Luo. Department of Mathematics Iowa State University
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1 Lecture 14 Eigenvalue Problems Songting Luo Department of Mathematics Iowa State University MATH 562 Numerical Analysis II Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH562 MATH 562 Numerical Analys 1 / 14
2 Outline 1 Eigenvalue Problems Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH562 MATH 562 Numerical Analys 2 / 14
3 Outline 1 Eigenvalue Problems Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH562 MATH 562 Numerical Analys 3 / 14
4 Eigenvalue and Eigenvectors Eigenvalue problem of m ˆ m matrix A is Ax λx with eigenvalues λ and eigenvectors x (nonzero) The set of all the eigenvalues of A is the spectrum of A Eigenvalue are generally used where a matrix is to be compounded iteratively Eigenvalues are useful for algorithmic and physical reasons Algorithmically, eigenvalue analysis can reduce a coupled system to a collection of scalar problems Physically, eigenvalue analysis can be used to study resonance and stability of physical systems Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH562 MATH 562 Numerical Analys 4 / 14
5 Eigenvalue Decomposition WEigenvalue decomposition of A is A XΛX 1 or AX XΛ 1 with eigenvectors x i as columns of X and eigenvalues λ i along diagonal of Λ. Alternatively, Ax i λ i x i Eigenvalue decomposition is change of basis to eigenvector coordinates Ax b Ñ px 1 bq ΛpX 1 xq Note that eigenvalue decomposition may not exist Question: How does eigenvalue decomposition differ from SVD? Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH562 MATH 562 Numerical Analys 5 / 14
6 Geometric Multiplicity Eigenvectors corresponding to a single eigenvalue λ form an eigenspace E λ Ď C mˆm Eigenspace is invariant in that AE λ Ď E λ Dimension of E λ is the maximum number of linearly independent eigenvectors that can be found Geometric multiplicity of λ is dimension of E λ, i.e., dimpnullpa λiqq Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH562 MATH 562 Numerical Analys 6 / 14
7 Algebraic Multiplicity The characteristic polynomial of A is degree m polynomial p A pzq detpzi Aq pz λ 1 qpz λ 2 q pz λ m q which is monic in that coefficient of z m is 1 λ is an eigenvalue of A iff p A pλq 0 Proof. Algebraic multiplicity of λ is its multiplicity as a root of p A Any matrix A has m eigenvalues, counted with algebraic multiplicity Question: What are the eigenvalues of a triangular matrix? Question: How are geometric multiplicity and algebraic multiplicity related? Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH562 MATH 562 Numerical Analys 7 / 14
8 Similarity Transformations The map A Ñ Y 1 AY is a similarity transformation of A for any nonsingular Y A and B are similar if there is a similarity transformation B Y 1 AY Theorem If Y is nonsingular, then A and Y 1 AY have the same characteristic polynomials, eigenvalues, and algebraic and geometric multiplicities. For characteristic polynomial: detpzi Y 1 AYq detpy 1 pzi AqYq detpzi Aq so algebraic multiplicities remain the same If x P E λ for A, then Y 1 x is in eigenspace of Y 1 AY corresponding to λ, and vice versa, so geometric multiplicities remain the same. Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH562 MATH 562 Numerical Analys 8 / 14
9 Algebraic Multiplicity ě Geometric Multiplicity Let n be be geometric multiplicity of λ for A. Let ˆV P C mˆn constitute of orthonormal basis of the E λ Extend ˆV to unitary V rˆv, Ṽs P Cmˆm and form «ff j ˆV AˆV ˆV AṼ λi C B V AV Ṽ AˆV Ṽ AṼ 0 D detpzi Bq detpzi λiqdetpzi λdq pz λq n detpzi λdq, so the algebraic multiplicity of λ as an eigenvalue of B is ě n A and B are similar, so the algebraic multiplicity of λ as an eigenvalue of A is at least ě n» fi» fi Examples: A 2 fl, B 2 1 fl Their 2 2 characteristic polynomial is pz 2q 3, so algebraic multiplicity of λ 2 is 3. But geometric multiplicity of A is 3 and that of B is 1. Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH562 MATH 562 Numerical Analys 9 / 14
10 Defective and Diagonalizable Matrices An eigenvalue of a matrix is defective if its algebraic multiplicity ą its geometric multiplicity A matrix is defective if it has a defective eigenvalue. Otherwise, it is called nondefective. Theorem An m ˆ m matrix A is nondefective iff it has an eigenvalue decomposition A XΛX 1 (ð) Λ is nondefective, and A is similar to Λ, so A is nondefective. (ñ) A nondefective matrix has m linearly independent eigenvectors. Take them as columns of X to obtain A XΛX 1 Nondefective matrices are therefore also said to be diagonalizable. Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH562 MATH 562 Numerical Analys 10 / 14
11 Determinant and Trace Determinant of A is detpaq ś m j 1 λ j, because detpaq p 1q m detp Aq p 1q m p A p0q Trace of A is trpaq ř m j 1 λ j, since p A pzq detpzi Aq z m p A pzq j 1 mź j 1 λ j mÿ a jj z m 1 ` Opz m 2 q j 1 mź mÿ pz λ j q z m λ j z m 1 ` Opz m 2 q j 1 Question: Are these results valid for defective or nondefective matrices? Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH562 MATH 562 Numerical Analys 11 / 14
12 Unitary Diagonalization A matrix A is unitarily diagonalizable if A QΛQ for a unitary matrix Q A hermitian matrix is unitarily diagonalizable, with real eigenvalues A matrix A is normal if A A AA Examples of normal matrices include hermitian matrices, skew hermitian matrices hermitian ô matrix is normal and all eigenvalues are real skew hermitian ô matrix is normal and all eigenvalues are imaginary If A is both triangular and normal, then A is diagonal Unitarily diagonalizable ô normal By induction using Schur factorization (next) Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH562 MATH 562 Numerical Analys 12 / 14
13 Schur Factorization Schur factorization is A QTQ, where Q is unitary and T is upper triangular Theorem Every square matrix A has a Schur factorization. Proof Proof by induction on dimension of A. Case m 1 is trivial. For m ě 2, let x be any unit eigenvector of A, with corresponding eigenvalue λ. Let Ujbe unitary matrix with x as first column. Then λ w U AU. By induction hypothesis, there is a Schur 0 C j factorization T 1 0 λ w V V CV. Let Q U, T 0 V 0 T then A QTQ j, and Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH562 MATH 562 Numerical Analys 13 / 14
14 Eigenvalue Revealing Factorization Eigenvalue-revealing factorization of square matrix A Diagonalization A XΛX 1 (nondefective A) Unitary Diagonalization A QΛQ (normal A) Unitary triangularization (Schur factorization) A QTQ (any A) Jordan normal form A XJX 1 where J block diagonal with» fi λ i 1. J i λ.. ffi i ffi ffi ffi fl... 1 λi In general, Schur factorization is used, because Unitary matrices are involved, so algorithm tends to be more stable If A is normal, then Schur form is diagonal Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH562 MATH 562 Numerical Analys 14 / 14
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