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11 Nov 2019
In this question, all norms in R" are with respect to the dot product. Let A be an n xn real symmetric matrix, and let P be an orthogonal matrix such that PT AP D, diagonal. (a) Show that if x E Rn and y = PTx, then xTAx = yTDy and llxll = llyll. (b) Suppose A is the greatest eigenvalue of A. Show that the maximum value of yTDy for y of norm 1 iSA. (Use question 6.) (c) Deduce that the maximum value of xTAx for x of norm 1 is λ.
In this question, all norms in R" are with respect to the dot product. Let A be an n xn real symmetric matrix, and let P be an orthogonal matrix such that PT AP D, diagonal. (a) Show that if x E Rn and y = PTx, then xTAx = yTDy and llxll = llyll. (b) Suppose A is the greatest eigenvalue of A. Show that the maximum value of yTDy for y of norm 1 iSA. (Use question 6.) (c) Deduce that the maximum value of xTAx for x of norm 1 is λ.
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