26-symmetric

26-symmetric - SYMMETRIC MATRICES Math 21b, O. Knill...

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Unformatted text preview: SYMMETRIC MATRICES Math 21b, O. Knill SYMMETRIC MATRICES. A matrix A with real entries is symmetric , if A T = A . EXAMPLES. A = 1 2 2 3 is symmetric, A = 1 1 3 is not symmetric. EIGENVALUES OF SYMMETRIC MATRICES. Symmetric matrices A have real eigenvalues. PROOF. The dot product is extend to complex vectors as ( v, w ) = ∑ i v i w i . For real vectors it satisfies ( v, w ) = v · w and has the property ( Av, w ) = ( v, A T w ) for real matrices A and ( λv, w ) = λ ( v, w ) as well as ( v, λw ) = λ ( v, w ). Now λ ( v, v ) = ( λv, v ) = ( Av, v ) = ( v, A T v ) = ( v, Av ) = ( v, λv ) = λ ( v, v ) shows that λ = λ because ( v, v ) 6 = 0 for v 6 = 0. EXAMPLE. A = p- q q p has eigenvalues p + iq which are real if and only if q = 0. EIGENVECTORS OF SYMMETRIC MATRICES. Symmetric matrices have an orthonormal eigenbasis PROOF. If Av = λv and Aw = μw . The relation λ ( v, w ) = ( λv, w ) = ( Av, w ) = ( v, A T w ) = ( v, Aw ) = ( v, μw ) = μ ( v, w ) is only possible if ( v, w ) = 0 if λ 6 = μ . WHY ARE SYMMETRIC MATRICES IMPORTANT? In applications, matrices are often symmetric. For ex- ample in geometry as generalized dot products v · Av , or in statistics as correlation matrices Cov[ X k , X l ] or in quantum mechanics as observables or in neural networks as learning maps x 7→ sign( W x ) or in graph theory as adjacency matrices etc. etc. Symmetric matrices play the same role as real numbers do among the complex numbers. Their eigenvalues often have physical or geometrical interpretations. One can also calculate with symmetric matrices like with numbers: for example, we can solve B 2 = A for B if A is symmetric matrix and B is square root of A .) This is not possible in general: try to find a matrix B such that B 2 = 1 ......
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This note was uploaded on 04/06/2008 for the course MATH 21B taught by Professor Judson during the Spring '03 term at Harvard.

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