Symmetric Matrices and Quadratic Forms
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1 7 Symmetric Matrices ad Quadratic Forms 7.1 DIAGONALIZAION OF SYMMERIC MARICES
2 SYMMERIC MARIX A symmetric matrix is a matrix A such that. A = A Such a matrix is ecessarily square. Its mai diagoal etries are arbitrary, but its other etries occur i pairso opposite sides of the mai diagoal. Slide 7.1-2
3 SYMMERIC MARIX heorem 1: If A is symmetric, the ay two eigevectors from differet eigespaces are orthogoal. Proof: Let v 1 ad v 2 be eigevectors that correspod to distict eigevalues, say, 1 ad 2. v v = 0 v v ( v ) v ( v ) v o show that, compute 1 2 = = A = (v A )v = v ( Av ) = v ( v ) = v v = λ v v Sice v 1 is a eigevector Sice A = A Sice v 2 is a eigevector Slide 7.1-3
4 SYMMERIC MARIX ( )v v = v v = Hece. But, so. A matrix A is said to be orthogoally diagoalizable if there are a orthogoal matrix P 1 (with ) ad a diagoal matrix D such that P = Such a diagoalizatio requires liearly idepedet ad orthoormal eigevectors. Whe is this possible? P A = PDP = PDP ----(1) If A is orthogoally diagoalizable as i (1), the A = ( PDP ) = P D P = PDP = A 1 Slide 7.1-4
5 SYMMERIC MARIX hus A is symmetric. heorem 2: A matrix A is orthogoally diagoalizable if ad oly if A is symmetric matrix. Example 1: Orthogoally diagoalize the matrix A = , whose characteristic equatio is = = ( 7) ( + 2) Slide 7.1-5
6 SYMMERIC MARIX Solutio: he usual calculatios produce bases for the eigespaces: 1 1/ 2 1 = 7 : v = 0,v = 1 ; = 2 : v = 1/ Although v 1 ad v 2 are liearly idepedet, they are ot orthogoal. v v v v v 2 1 he projectio of v 2 oto v 1 is Slide 7.1-6
7 SYMMERIC MARIX he compoet of v 2 orthogoal to v 1 is 1/ 2 1 1/ 4 v v 1/ z = v v = 1 0 = v v / 4 he {v 1, z 2 } is a orthogoal set i the eigespace for. = 7 (Note that z 2 is liear combiatio of the eigevectors v 1 ad v 2, so z 2 is i the eigespace). Slide 7.1-7
8 SYMMERIC MARIX Sice the eigespace is two-dimesioal (with basis v 1, v 2 ), the orthogoal set {v 1, z 2 } is a orthogoal basis for the eigespace, by the Basis heorem. Normalize v 1 ad z 2 to obtai the followig orthoormal basis for the eigespace for : 1/ 2 1/ 18 u = 0,u 4 / 18 1 = 2 1/ 2 1/ 18 = 7 Slide 7.1-8
9 SYMMERIC MARIX A orthoormal basis for the eigespace for 2 2 / u = 2v = 1 = 1/ v / 3 = 2 is By heorem 1, u 3 is orthogoal to the other eigevectors u 1 ad u 2. Hece {u 1, u 2, u 3 } is a orthoormal set. Slide 7.1-9
10 SYMMERIC MARIX Let 1/ 2 1/ 18 2 / P = [ u u u ] = 0 4 / 18 1/ 3, D = 1/ 2 1/ 18 2 / A = PDP 1 he P orthogoally diagoalizes A, ad. Slide
11 HE SPECRAL HEOREM he set if eigevalues of a matrix A is sometimes called the spectrum of A, ad the followig descriptio of the eigevalues is called a spectral theorem. heorem 3: he Spectral heorem for Symmetric Matrices A symmetric matrix A has the followig properties: a. A has real eigevalues, coutig multiplicities. Slide
12 HE SPECRAL HEOREM b. he dimesio of the eigespace for each eigevalue equals the multiplicity of as a root of the characteristic equatio. c. he eigespaces are mutually orthogoal, i the sese that eigevectors correspodig to differet eigevalues are orthogoal. d. A is orthogoally diagoalizable. Slide
13 SPECRAL DECOMPOSIION Suppose, where the colums of P are orthoormal eigevectors u 1,,u of A ad the correspodig eigevalues 1,, are i the diagoal matrix D. A = PDP 1 P = P 1 he, sice, 0 1 u1 [ u u ] A = PDP = L O 1 M 0 u = [ u L u ] 1 1 u M u 1 Slide
14 SPECRAL DECOMPOSIION Usig the colum-row expasio of a product, we ca write A = u u + u u + L+ u u (2) his represetatio of A is called a spectral decompositio of A because it breaks up A ito pieces determied by the spectrum (eigevalues) of A. Each term i (2) is a matrix of rak 1. u u For example, every colum of is a multiple of u 1. u u Each matrix is a projectio matrix i the sese that j j for each x i, the vector (u u )x is the orthogoal j j projectio of x oto the subspace spaed by u j. Slide
15 SPECRAL DECOMPOSIION Example 2: Costruct a spectral decompositio of the matrix A that has the orthogoal diagoalizatio / 5 1/ / 5 1/ 5 A = 2 4 = 1/ 5 2 / / 5 2 / 5 Solutio: Deote the colums of P by u 1 ad u 2. he A = 8u u + 3u u Slide
16 SPECRAL DECOMPOSIION o verify the decompositio of A, compute 2 / 5 4 / 5 2 / 5 u u = 2 / 5 1/ = 1/ 5 2 / 5 1/ 5 1/ 5 1/ 5 2 / 5 u u = 1/ 5 2 / / 5 = 2 / 5 4 / 5 ad 32 / 5 16 / 5 3/ 5 6 / u u + 3u u = / 5 8 / 5 + = = 6 / 5 12 / A Slide
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