Lecture 33 True False Questions True Every eigenvalue has an eigenvector det A I 0 A I is singular so we can solve A I x 0 True Eigenvectors from different eigenvalues are independent S AS True Positive de nite matrices have full rank n Because this is positive de nite all eigenvalues are nonnegative meaning there are no zero eigenvalues If x x x are independent vectors you can think of this as our basis then v c x c x c x How do we nd the values of c We have to consider two cases The vectors x are not orthogonal The vectors x are orthogonal Handout 1 Note upper triangular matrices have the eigenvalues on their diagonal The eigenvalues of A are 0 1 3 The eigenvalues of A are 0 1 3 The eigenvalues of e are e The eigenvectors of A are 1 0 0 1 3 0 1 3 1 S is the matrix of eigenvectors is the matrix with eigenvalues on its diagonal We can solve and Practice nding eigenvectors Review nding the inverse of a 3x3 diagonal matrix Which matrices have orthogonal eigenvectors Symmetric matrices 2 Start by nding the eigenvalues and eigenvectors We know that one of eigenvalues is one because that is a given for Markov matrices We know that another eigenvalue is zero because row one and row two add up to row three the rows are dependent And we know the last eigenvalues is 2 because the three eigenvalues have to add up to the trace Equation for the state at time n We know that goes to zero everywhere except where the eigenvalue is one so our only eigenvalue of interest is the corresponding eigenvector x 5 7 12 We know that the steady state is a multiple of the eigenvector 5 7 12 3 The eigenvalues of A are the same as the eigenvalues of The eigenvalues of A are the same as the eigenvalues of The eigenvalues of A are not the same as the eigenvalues of 4 Yes it is positive de nite No it is not positive de nite Yes it is positive de nite No Yes 5 Principal component analysis 6 Not relevant to the exam
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