SECTIONS 5.2/5.4 BASIC PROPERTIES OF EIGENVALUES AND EIGENVECTORS / SIMILARITY TRANSFORMATIONS

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1 SECINS 5/54 BSIC PRPERIES F EIGENVUES ND EIGENVECRS / SIMIRIY RNSFRMINS Eigenvalues of an n : there exists a vector x for which x = x Such a vector x is called an eigenvector, and (, x) is called an eigenpair of n matrix are { } Some applications -- see Section 5 Clearly any nonzero scalar multiple of an eigenvector is also an eigenvector: x = x ( αx) = ( αx) () () ( ) More generally, if x, x, K, x are eigenvectors associated with the same eigenvalue, then any nonzero linear combination of them is also an eigenvector: () () ( ) ( α x + α x + + α x ) () () = α( x ) + α ( x ) + + α ( x () () ( ) = ( α x + α x + + α x ) ( ) ) he eigenvalues are the n roots of the characteristic equation det( I) = his is a polynomial of degree n in the variable EXMPE 8 = det( I) = det =

2 he eigenvalues are the zeros of this cubic polynomial:, + 4i, 4i For each, an eigenvector can be obtained from the homogeneous system of linear equations ( I) x = For example, for =, x x 8 x 3 = Note that the coefficient matrix of this system is necessarily singular It can be solved by choosing an arbitrary value for one of the variables, and then solving for the other variables For example, letting x, gives the system 7x 8x 3 = 4x x 5x = 5 + 3x = = 8 which gives x = NE: Real matrices can have complex eigenvalues ny complex eigenvalues occur in complex conjugate pairs Eigenvectors associated with complex eigenvalues are in general complex valued Matrices and B are said to be similar if there exists a nonsingular matrix such that = B Similar matrices have the same eigenvalues, as their characteristic polynomials are identical: det( I) = det( B I) = = det( ( B I) ) ( det ) det( B I) ( det ) = det( B I), since det = det he eigenvectors of are easily obtained from those of B by multiplying by : 83

3 Bx = x B x = ( x) ( x) = ( x) HER SIMPE EIGENVUE/EIGENVECR REINSHIPS (i) eigenvalues and eigenvectors of a shifted matrix αi : x = x ( αi ) x = ( α) x (ii) is nonsingular if and only if is not an eigenvalue of -- follows since is an eigenvalue if and only if there exists a nonzero vector such that x = (iii) eigenvalues and eigenvectors of : x = x x = x (iv) he eigenvalues of and are the same (as their characteristic polynomials are the same), but there is no simple relationship between their eigenvectors (v) eigenvalues and eigenvectors of powers of : x = x x = ( x) = ( x) = x In general, the eigenvalues of those of are, and the eigenvectors are the same as (vi) he eigenvalues of a triangular matrix are clearly its diagonal entries he eigenvalues of a bloc triangular matrix = M are the union of the eigenvalues of the diagonal submatrices,,, K How many linearly independent eigenvectors can an n n matrix have? nswer: anywhere from to n he following examples illustrate the possible cases when n = 3 84

4 has 3 linearly independent eigenvectors has linearly independent eigenvectors has linearly independent eigenvector matrix is called defective (or not semisimple in your textboo) if it does not have a set of n linearly independent eigenvectors matrix with n linearly independent eigenvectors is nondefective or (in our text) semisimple NE: If a matrix has an eigenvalue of multiplicity greater than, it may be defective or nondefective: see the above 3 examples (where each matrix has one eigenvalue of multiplicity 3) HEREM 5 (page 38) Eigenvectors corresponding to distinct eigenvalues are linearly independent CRRY 5 If has n distinct eigenvalues, then is nondefective HEREM 546 (page 335) matrix is nondefective if and only if there exists a nonsingular matrix V such that V V = D, where D = n and {,,, } K n are the eigenvalues of 85

5 Proof et that () () ( n) x, x, K, x be a set of n linearly independent eigenvectors, and suppose ( ) () ( n) et V [ x x x ] x ( i) ( i) = x, i n () i = hen the n equations in () are equivalent to the matrix equation V = VD, which implies that V V = D For the converse, if there exists a matrix V such that V V = matrix equation is equivalent to the n equations D, then V = VD his x ( i) ( i) = x, i n, i (i) where x denotes the i-th column vector of V Since V is nonsingular, the n () () ( n) eigenvectors x, x, K, x are linearly independent, implying that is nondefective NE: if matrix V diagonalizes (that is, V V = D is diagonal), then the above proof shows that the column vectors of V are a set of n linearly independent eigenvectors of Definitions is a real symmetric matrix if = complex matrix is Hermitian if = U is a (real) orthogonal matrix if U U = UU = I or U = U U is a (complex) unitary matrix if U U = UU = I or U = U and B are similar if = B and B are orthogonally similar if = U BU, where U is orthogonal and B are unitarily similar if = U BU, where U is unitary HEREM 54 (Schur's heorem, page 337) Given any square matrix, there exists a unitary matrix U such that = U U = U U is upper triangular he diagonal entries of are the eigenvalues of 86

6 HEREM 54 (Spectral heorem, page 339) If is a Hermitian matrix, then there exists a unitary matrix U such that D = U U = U U is diagonal he diagonal entries of D are the eigenvalues of Proof By Schur's heorem, U U is upper triangular But ( U U ) = U U = U U implies that U U is also Hermitian, and thus is diagonal NE: If is real symmetric, then the matrix U of heorem 54 is a (real) orthogonal matrix such that U U is diagonal (see heorem 549, page 34) It follows from the Spectral heorem that the eigenvalues of a Hermitian matrix (including a real symmetric matrix) are real: since U U is Hermitian and diagonal, its diagonal entries (which are the eigenvalues of ) are real See Corollary 543, page 339 he most important result for arbitrary real matrices using (real) orthogonal similarity transformations is HEREM 54 (the Real Schur heorem or the Wintner-Murnaghan heorem, page 34) If is a real matrix, then there exists a (real) orthogonal matrix U such that = U U is a quasi-triangular matrix = M (hat is, is bloc upper triangular and each main diagonal submatrix ii is either or ) NE: the eigenvalues of are the union of the eigenvalues of the main diagonal submatrices ii ny real eigenvalues of will occur as submatrices and any pairs of complex conjugate eigenvalues of will occur as the eigenvalues of some submatrix 87

7 SUMMRY F RESUS USING SIMIRIY RNSFRMINS GENER SIMIRIY RNSFRMINS is nondefective (semisimple) is defective (not semisimple) -- there exists V such -- there exists X such that V V is diagonal that X X is bidiagonal (the Jordan normal form) UNIRY SIMIRIY RNSFRMINS arbitrary matrix matrix is Hermitian -- there exists a unitary -- there exists a unitary matrix U such that U U matrix U such that U U is upper triangular is diagonal RHGN SIMIRIY RNSFRMINS arbitrary real matrix is real symmetric -- there exists an -- there exists an orthogonal matrix P orthogonal matrix P such that P P is such that P P is quasi-triangular (with all diagonal diagonal blocs of order or ) he equivalence of computing eigenvalues and computing the zeros of polynomials We have already seen that the eigenvalues are the zeros of the characteristic polynomnial n the other hand, given any (monic) polynomial p( n ) = a + a + a + + a n + n 88

8 89 the companion matrix of ) ( p is = a a a a n n, where all unspecified entries of are hen ) ( ) det( p I = and thus the zeros of ) ( p are the eigenvalues of