16 - Symmetric Matrices

16 - Symmetric Matrices - 1 Math 1b Practical March 02,...

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1 Math 1b Practical — March 02, 2011 Real symmetric matrices Amatrix A is symmetric when A > = A . In the following A will denote the conjugate- transpose of A , the matrix whose entries are the complex conjugates of the entries of A > . A (complex) matrix is called Hermitian when A = A . There are results about Hermitian matrices that are similar to those about real symmetric matrices proved here. See e.g. Section 2 of Chapter 6 of LADW. Theorem 1. The eigenvalues of a real symmetric matrix are real. Proof: Let A be real symmetric and suppose A x = λ x where λ and x are possibly complex, and x 6 = 0 . We take the conjugate tranpose to get λ x =( λ x ) A x ) = x A = x A, the latter equality because A is symmetric and real. Then λ x x x A ) x = x ( A x )= λ x x . Now x x is nonzero (it is real too; it is what we called || x 2 for complex x ). So we cancel to Fnd λ = λ , which means that λ is real. ut Remark. We have seen examples where eigenvectors have complex numbers as coordi- nates. But if λ is a real eigenvalue of a real matrix A , then we can always Fnd a real eigenvector corresponding to λ . A reason is that the row operations we can use to Fnd a basis for the null space of A λI produce real vectors. Exercise. All eigenvalues of a Hermitian matrix are real. Theorem 2. If x and y are eigenvectors of a real symmetric matrix A corresponding to distinct eigenvalues λ and μ ,then h x , y i =0 . Think of both x and y as column vectors, as usual. Then h x , y i = x > y .W ehav e A x = λ x , A y = μ y .Then x > A > = x > A = λ x > ,and λ x > y x > A ) y = x > ( A y μ x > y . Since λ 6 = μ ,wemusthave x > y =0. ut Theorem 3. A real symmetric matrix is diagonalizable. In fact, R n has an orthonormal basis of eigenvectors for any real symmetric matrix S . If these are taken as the columns of an (orthogonal) matrix E E > SE is a diagonal matrix with the eigenvalues of S on the diagonal.
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This note was uploaded on 11/10/2011 for the course MA 1B taught by Professor Aschbacher during the Winter '08 term at Caltech.

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16 - Symmetric Matrices - 1 Math 1b Practical March 02,...

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