7 Symmetric Matrices and Quadratic Forms 7.1 DIAGONALIZAION OF SYMMERIC MARICES
SYMMERIC MARIX A symmetric matrix is a matrix A such that. A A Such a matrix is necessarily square. Its main diagonal entries are arbitrary, but its other entries occur in pairs on opposite sides of the main diagonal. Slide 7.1-2
SYMMERIC MARIX heorem 1: If A is symmetric, then any two eigenvectors from different eigenspaces are orthogonal. Proof: Let v 1 and v 2 be eigenvectors that correspond to distinct eigenvalues, say, λ 1 and λ 2. v v 0 λ v v (λ v ) v ( v ) v o show that, compute 1 2 A 1 1 2 1 1 2 1 2 (v A )v v ( Av ) 1 2 1 2 v(λ v ) 1 2 2 λ v v v v 2 1 2 2 1 2 Since v 1 is an eigenvector Since A A Since v 2 is an eigenvector Slide 7.1-3
SYMMERIC MARIX (λ λ )v v 0 1 2 1 2 λ λ 0 v v 0 1 2 1 2 nn Hence. But, so. An matrix A is said to be orthogonally diagonalizable if there are an orthogonal matrix P 1 (with ) and a diagonal matrix D such that P Such a diagonalization requires n linearly independent and orthonormal eigenvectors. When is this possible? P A PDP PDP ----(1) If A is orthogonally diagonalizable as in (1), then A ( PDP ) P D P PDP A 1 Slide 7.1-4
SYMMERIC MARIX hus A is symmetric. nn heorem 2: An matrix A is orthogonally diagonalizable if and only if A is symmetric matrix. Example 1: Orthogonally diagonalize the matrix 3 2 4 A 2 6 2 4 2 3, whose characteristic equation is 3 2 2 0 λ 12λ 21λ 98 (λ 7) (λ 2) Slide 7.1-5
SYMMERIC MARIX Solution: he usual calculations produce bases for the eigenspaces: 1 1/ 2 1 λ 7 : v 0,v 1 ;λ 2 : v 1/ 2 1 2 3 1 0 1 Although v 1 and v 2 are linearly independent, they are not orthogonal. v v v v v 2 1 he projection of v 2 onto v 1 is. 1 1 1 Slide 7.1-6
SYMMERIC MARIX he component of v 2 orthogonal to v 1 is 1/ 2 1 1/ 4 v v 1/ 2 2 1 z v v 1 0 1 2 2 1 v v 2 1 1 0 1 1/ 4 hen {v 1, z 2 } is an orthogonal set in the eigenspace for. λ 7 (Note that z 2 is linear combination of the eigenvectors v 1 and v 2, so z 2 is in the eigenspace). Slide 7.1-7
SYMMERIC MARIX Since the eigenspace is two-dimensional (with basis v 1, v 2 ), the orthogonal set {v 1, z 2 } is an orthogonal basis for the eigenspace, by the Basis heorem. Normalize v 1 and z 2 to obtain the following orthonormal basis for the eigenspace for : 1/ 2 1/ 18 u 0,u 4 / 18 1 2 1/ 2 1/ 18 λ 7 Slide 7.1-8
SYMMERIC MARIX An orthonormal basis for the eigenspace for 2 2 / 3 1 1 u 2v 1 1/ 3 3 3 2v 3 3 2 2 / 3 λ2 is By heorem 1, u 3 is orthogonal to the other eigenvectors u 1 and u 2. Hence {u 1, u 2, u 3 } is an orthonormal set. Slide 7.1-9
SYMMERIC MARIX Let 1/ 2 1/ 18 2 / 3 7 0 0 P u u u 0 4 / 18 1/ 3, D 0 7 0 1 2 3 1/ 2 1/ 18 2 / 3 0 0 2 A PDP 1 hen P orthogonally diagonalizes A, and. Slide 7.1-10
HE SPECRAL HEOREM he set if eigenvalues of a matrix A is sometimes called the spectrum of A, and the following description of the eigenvalues is called a spectral theorem. heorem 3: he Spectral heorem for Symmetric Matrices nn An symmetric matrix A has the following properties: a. A has n real eigenvalues, counting multiplicities. Slide 7.1-11
HE SPECRAL HEOREM b. he dimension of the eigenspace for each eigenvalue λ equals the multiplicity of λ as a root of the characteristic equation. c. he eigenspaces are mutually orthogonal, in the sense that eigenvectors corresponding to different eigenvalues are orthogonal. d. A is orthogonally diagonalizable. Slide 7.1-12
SPECRAL DECOMPOSIION Suppose, where the columns of P are orthonormal eigenvectors u 1,,u n of A and the corresponding eigenvalues λ 1,,λ n are in the diagonal matrix D. 1 hen, since, A A PDP 1 PDP λ u 1 1 P u 1 P λ u n n u u u n 1 n λ 0 u 1 1 0 λ u n n Slide 7.1-13
SPECRAL DECOMPOSIION Using the column-row expansion of a product, we can write A λ u u λ u u λ u u 1 1 1 2 2 2 n n n ----(2) his representation of A is called a spectral decomposition of A because it breaks up A into pieces determined by the spectrum (eigenvalues) of A. nn Each term in (2) is an matrix of rank 1. λ u u For example, every column of is a multiple of u 1. uu 1 1 1 Each matrix is a projection matrix in the sense that j j n for each x in, the vector (u u )x is the orthogonal j j projection of x onto the subspace spanned by u j. Slide 7.1-14
SPECRAL DECOMPOSIION Example 2: Construct a spectral decomposition of the matrix A that has the orthogonal diagonalization 7 2 2 / 5 1/ 5 8 0 2 / 5 1/ 5 A 2 4 1/ 5 2 / 5 0 3 1/ 5 2 / 5 Solution: Denote the columns of P by u 1 and u 2. hen A 8u u 3u u 1 1 2 2 Slide 7.1-15
SPECRAL DECOMPOSIION o verify the decomposition of A, compute 2 / 5 4 / 5 2 / 5 u u 2 / 5 1/ 5 1 1 1/ 5 2 / 5 1/ 5 1/ 5 1/ 5 2 / 5 u u 1/ 5 2 / 5 2 2 2 / 5 2 / 5 4 / 5 and 32 / 5 16 / 5 3 / 5 6 / 5 7 2 8u u 3u u 1 1 2 2 16 / 5 8 / 5 6 / 5 12 / 5 2 4 A Slide 7.1-16