Denote by Sax the vector space of symmetric n x n matrices and observe that ATA...

 

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Denote by Sax the vector space of symmetric n x n matrices and observe that ATA € S*** VA € R***, where the superscript T denotes transpose. So, define f: Rxn Saxn by f(A) := ATA. (a) Explain why f is differentiable and, for HER*, calculate the directional derivative Of(A). Hint. Follow the procedure of Remark 4.5.3. For the rest of this question, assume that k = n, i.e., A is a square n x n matrix and let I,, be the nxn identity matrix. Define O(n. R):= f(1) = {0 ROTO = 1}. Recall that, since f is differentiable, Df(A) is the linear map from Rx to Sx" defined by Df(A)H: дн f(A). (b) Write down the equation satisfied by the matrices in ker (Df(n)) and write down a basis for ker (Df(1)). Hint, ker (Df(n)) = {HER"x": Ouf(In)=0}. So write down the equation Of(In) = 0 explicitly as an equation for H and then write down matrices that will form a basis of the space of solutions of this equation. The case n = 2 is the easiest to analyse. (c) Show that Df(In) is surjective. Hint. For each S Sax", you have to find HER*** (in terms of S) such that Juf(1) = S. The choice of H is not unique; H can be changed by any element of ker (Df(I)). (d) Show, more generally, that Df(O) is surjective at all matrices O = O(n, R). Hint. Again, for each S involves O as well as S. Sx", you have to find H€ Rx" such that Of(0)) = S; H now Denote by Sax the vector space of symmetric n x n matrices and observe that ATA € S*** VA € R***, where the superscript T denotes transpose. So, define f: Rxn Saxn by f(A) := ATA. (a) Explain why f is differentiable and, for HER*, calculate the directional derivative Of(A). Hint. Follow the procedure of Remark 4.5.3. For the rest of this question, assume that k = n, i.e., A is a square n x n matrix and let I,, be the nxn identity matrix. Define O(n. R):= f(1) = {0 ROTO = 1}. Recall that, since f is differentiable, Df(A) is the linear map from Rx to Sx" defined by Df(A)H: дн f(A). (b) Write down the equation satisfied by the matrices in ker (Df(n)) and write down a basis for ker (Df(1)). Hint, ker (Df(n)) = {HER"x": Ouf(In)=0}. So write down the equation Of(In) = 0 explicitly as an equation for H and then write down matrices that will form a basis of the space of solutions of this equation. The case n = 2 is the easiest to analyse. (c) Show that Df(In) is surjective. Hint. For each S Sax", you have to find HER*** (in terms of S) such that Juf(1) = S. The choice of H is not unique; H can be changed by any element of ker (Df(I)). (d) Show, more generally, that Df(O) is surjective at all matrices O = O(n, R). Hint. Again, for each S involves O as well as S. Sx", you have to find H€ Rx" such that Of(0)) = S; H now