Gradient of the Rayleigh Quotient For a symmetric matrix A and a nonzero vector , the...

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Gradient of the Rayleigh Quotient For a symmetric matrix A and a nonzero vector , the Rayleigh Quotient is defined by xT Ax xTx In the case where is an eigenvector, then the Rayleigh Quotient will approximate \, the eigenvalue corresponding to that eigenvector. 1. Show that VxTx = 2x. 2. Show that Vx Ax = (A+AT) x = 2Ax. 3. Using part 1 and part 2, show that T Ax xTx xT AxI 2x A |||| 4. Let vi denote the ith eigenvector of A and denote its corresponding eigenvalue. Show that xT Ax = 0. xTx x=vi Be sure to show all your work and provide justifications for every step to receive full credit. Gradient of the Rayleigh Quotient For a symmetric matrix A and a nonzero vector , the Rayleigh Quotient is defined by xT Ax xTx In the case where is an eigenvector, then the Rayleigh Quotient will approximate \, the eigenvalue corresponding to that eigenvector. 1. Show that VxTx = 2x. 2. Show that Vx Ax = (A+AT) x = 2Ax. 3. Using part 1 and part 2, show that T Ax xTx xT AxI 2x A |||| 4. Let vi denote the ith eigenvector of A and denote its corresponding eigenvalue. Show that xT Ax = 0. xTx x=vi Be sure to show all your work and provide justifications for every step to receive full credit.