# u7 - Unit VII Symmetric Matrices 1 Spectral Theorem A...

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Unit VII: Symmetric Matrices1. Spectral TheoremAlinear transformation on a subspaceVofRnis a functionTfromVtoVthat satisfiesT(ax+by) =aT(x) +bT(y),x,yV, a, bscalars.Thematrix ofTwith respect to the basis{v1, . . . ,vm}ofVis them×mmatrixAsatisfyingT(vk) =mXj=1Ajkvj,1km.These definitions are virtually the same as those given earlier for a linear transformation onRn. Again note that thekth column ofAcontains the coordinates ofT(vk) with respect tothe basis{v1, . . . ,vm}. Also, note that ifc1, . . . , cmare the components ofuVwith respectto the basis{v1, . . . ,vm}, andd1, . . . , dmare the components ofT(u), thendj=kAjkck.LemmaIf{u1, . . . ,um}is an orthonormal basis forV, then the matrixAforTwithrespect to the basis is given byAjk=T(uk)·uj,1km.Proof.This follows fromT(uk)·uj=rArkur·uj=Ajk.Definition: Asymmetric linear transformation on a subspaceVofRnis a linear trans-formationTonVthat satisfiesT(x)·y=x·T(y),x,yV.Lemma.LetTbe a linear transformation on a subspaceVofRn, and let{u1, . . . ,um}be an orthonormal basis forV. ThenTis a symmetric linear transformation if and only ifthe matrix ofTwith respect to the basis{u1, . . . ,um}is a symmetric matrix.Proof.This follows fromAjk=T(uk)·uj=uk·T(uj) =T(uj)·uk=Akj.Theorem 1.LetAbe a (real) symmetricn×nmatrix. Then the roots of the character-istic polynomialpA(λ) = det (λI-A)ofAare real. ThusAhasnreal eigenvalues (countingmultiplicity).Proofsketch.Letλbe a root of the characteristic polynomial ofA. Then there is a nonzerovectorz= (z1, . . . , zn)Cnthat satisfies (λI-A)z=0. We express eachzj=xj+iyjas the sum of its real and imaginary parts. Thenz=x+iy, wherex= (x1, . . . , xn) andy= (y1, . . . , yn) are vectors inRn. We denotex-iy=z. Then fromzjzj=|zj|2, we obtainz·z=|zj|26= 0. SinceAz=λz, we haveλX|zj|2=Xjλzjzj=Xj(Az)jzj=Xj,kAkjzkzj.1
Since the matrixAis symmetric and has real entries, this is equal toXj,kzkAjkzj=Xkzk(Az)k=Xkzkλzk=λXk|zk|2.Dividing by|zj|2, we conclude thatλ=λ, andλis real.Theorem 2.IfTis a symmetric linear transformation, then eigenvectors ofTcorre-sponding to different eigenvalues are orthogonal.Proof.IfT(u) =λuandT(v) =μv, thenλu·v=T(u)·v=u·T(v) =μu·v. Thuseitherλ=μoru·v= 0.Theorem 3.LetTbe a symmetric linear transformation onRn. IfVis a subspace ofRnsuch thatT(V)V, thenT(V)V.Proof.LetwV. IfvV, thenT(v)V, so 0 =T(v)·w=v·T(w). Since thisholds for allvV, we obtainwV.Theorem 4 (Spectral Theorem).LetTbe a symmetric linear transformation onRn.Then there is an orthonormal basis forRnconsisting of eigenvectors ofT.