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

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Unformatted text preview: Unit VII: Symmetric Matrices 1. Spectral Theorem A linear transformation on a subspace V of R n is a function T from V to V that satisfies T ( a x + b y ) = aT ( x ) + bT ( y ) , x , y V, a,b scalars . The matrix of T with respect to the basis { v 1 ,..., v m } of V is the m m matrix A satisfying T ( v k ) = m X j =1 A jk v j , 1 k m. These definitions are virtually the same as those given earlier for a linear transformation on R n . Again note that the k th column of A contains the coordinates of T ( v k ) with respect to the basis { v 1 ,..., v m } . Also, note that if c 1 ,...,c m are the components of u V with respect to the basis { v 1 ,..., v m } , and d 1 ,...,d m are the components of T ( u ), then d j = k A jk c k . Lemma If { u 1 ,..., u m } is an orthonormal basis for V , then the matrix A for T with respect to the basis is given by A jk = T ( u k ) u j , 1 k m. Proof. This follows from T ( u k ) u j = r A rk u r u j = A jk . Definition: A symmetric linear transformation on a subspace V of R n is a linear trans- formation T on V that satisfies T ( x ) y = x T ( y ) , x , y V. Lemma. Let T be a linear transformation on a subspace V of R n , and let { u 1 ,..., u m } be an orthonormal basis for V . Then T is a symmetric linear transformation if and only if the matrix of T with respect to the basis { u 1 ,..., u m } is a symmetric matrix. Proof. This follows from A jk = T ( u k ) u j = u k T ( u j ) = T ( u j ) u k = A kj . Theorem 1. Let A be a (real) symmetric n n matrix. Then the roots of the character- istic polynomial p A ( ) = det ( I- A ) of A are real. Thus A has n real eigenvalues (counting multiplicity). Proofsketch. Let be a root of the characteristic polynomial of A . Then there is a nonzero vector z = ( z 1 ,...,z n ) C n that satisfies ( I- A ) z = . We express each z j = x j + iy j as the sum of its real and imaginary parts. Then z = x + i y , where x = ( x 1 ,...,x n ) and y = ( y 1 ,...,y n ) are vectors in R n . We denote x- i y = z . Then from z j z j = | z j | 2 , we obtain z z = | z j | 2 6 = 0. Since A z = z , we have X | z j | 2 = X j z j z j = X j ( Az ) j z j = X j,k A kj z k z j . 1 Since the matrix A is symmetric and has real entries, this is equal to X j,k z k A jk z j = X k z k ( Az ) k = X k z k z k = X k | z k | 2 . Dividing by | z j | 2 , we conclude that = , and is real. Theorem 2. If T is a symmetric linear transformation, then eigenvectors of T corre- sponding to different eigenvalues are orthogonal. Proof. If T ( u ) = u and T ( v ) = v , then u v = T ( u ) v = u T ( v ) = u v . Thus either = or u v = 0....
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This note was uploaded on 06/25/2008 for the course MATH 33a taught by Professor Lee during the Spring '08 term at UCLA.

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

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