SYMMETRIC MATRICES Math 21b O Knill SYMMETRIC MATRICES A matrix A with real entries is symmetric if AT A EXAMPLES A 1 2 2 3 is symmetric A 1 1 0 3 is not symmetric EIGENVALUES OF SYMMETRIC MATRICES Symmetric matrices A have real eigenvalues P PROOF The dot product is extend to complex vectors as v w i v i wi For real vectors it satisfies v w v w and has the property Av w v AT w for real matrices A and v w v w as well as v w v w Now v v v v Av v v AT v v Av v v v v shows that 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 v and Aw w The relation v w v w Av w v A T w v Aw v w v w is only possible if v w 0 if 6 WHY ARE SYMMETRIC MATRICES IMPORTANT In applications matrices are often symmetric For example in geometry as generalized dot products v Av or in statistics as correlation matrices Cov X k Xl 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 0 1 2 and B is square root of A This is not possible in general try to find a matrix B such that B 0 0 RECALL We have seen when an eigenbasis exists a matrix A can be transformed to a diagonal matrix B S 1 AS where S v1 vn The matrices A and B are similar B is called the diagonalization of A Similar matrices have the same characteristic polynomial det B det S 1 A S det A and have therefore the same determinant trace and eigenvalues Physicists call the set of eigenvalues also the spectrum They say that these matrices are isospectral The spectrum is what you see etymologically the name origins from the fact that in quantum mechanics the spectrum of radiation can be associated with eigenvalues of matrices SPECTRAL THEOREM Symmetric matrices A can be diagonalized B S 1 AS with an orthogonal S PROOF If all eigenvalues are different there is an eigenbasis and diagonalization is possible The eigenvectors are all orthogonal and B S 1 AS is diagonal containing the eigenvalues In general we can change the matrix A to A A C A t where C is a matrix with pairwise different eigenvalues Then the eigenvalues are different for all except finitely many t The orthogonal matrices St converges for t 0 to an orthogonal matrix S and S diagonalizes A WAIT A SECOND Why could we not perturb a general matrix At to have disjoint eigenvalues and At could be diagonalized St 1 At St Bt The problem is that St might become singular for t 0 See problem 5 first practice exam a b b a has the eigenvalues a b a b and the eigenvectors v1 EXAMPLE 1 The matrix A 1 and v2 2 They are orthogonal The orthogonal matrix S v1 v2 diagonalized A 1 1 1 2 1 1 1 EXAMPLE 2 The 3 3 matrix A 1 1 1 has 2 eigenvalues 0 to the eigenvectors 1 1 0 1 1 1 1 0 1 and one eigenvalue 3 to the eigenvector 1 1 1 All these vectors can be made orthogonal and a diagonalization is possible even so the eigenvalues have multiplicities SQUARE ROOT OF A MATRIX How do we find a square root of a given symmetric matrix Because S 1 AS B is diagonal and we know how to take a square root of the diagonal matrix B we can form C S BS 1 which satisfies C 2 S BS 1 S BS 1 SBS 1 A RAYLEIGH FORMULA We write also v w v w If v t is an eigenvector of length 1 to the eigenvalue t of a symmetric matrix A t which depends on t differentiation of A t t v t 0 with respect to t gives A0 0 v A v 0 0 The symmetry of A implies 0 v A0 0 v v A v 0 v A0 0 v We see that the Rayleigh quotient 0 A0 v v is a polynomial in t if A t only involves terms t t2 tm The 1 t2 formula shows how t changes when t varies For example A t has for t 2 the eigenvector t2 1 0 4 v v 4 Indeed v 1 1 2 to the eigenvalue 5 The formula tells that 0 2 A0 2 v v 4 0 t 1 t2 has at t 2 the derivative 2t 4 EXHIBITION Where do symmetric matrices occur Some informal motivation I PHYSICS In quantum mechanics a system is described with a vector v t which depends on time t The evolution is given by the Schroedinger equation v ih Lv where L is a symmetric matrix and h is a small number called the Planck constant As for any linear differential equation one has v t e ih Lt v 0 If v 0 is an eigenvector to the eigenvalue then v t eith v 0 Physical observables are given by symmetric matrices too The matrix L represents the energy Given v t the value of the observable A t is v t Av t For example if v is an eigenvector to an eigenvalue of the energy matrix L then the energy of v t is This is called the Heisenberg picture In order that v A t v v t Av t S t v AS t v we have A t S T AS t where S S T is the correct generalization of the adjoint to complex matrices S t satisfies S t S t 1 which is called unitary and the complex analogue of orthogonal The matrix A t S t AS t has the same eigenvalues as A and is similar to A II CHEMISTRY The adjacency matrix A of a graph with n vertices determines the graph one has A ij 1 if the two vertices i j are connected and zero otherwise The matrix A is symmetric The eigenvalues j are real and can be used to analyze the graph One interesting question is to what extent the eigenvalues determine the graph In chemistry one is interested in such problems because it allows to make rough computations of the electron density distribution of molecules In this so called Hu ckel theory the molecule is represented as a graph The eigenvalues j of that graph approximate the energies an electron …
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