Q:3 - 5 marks. Question 3. (5 PTS) Let T: Mat2,2(RR) - Mat2,2(RR) be defined as T(A) = AT. Recall that we have calculated its eigenvalues and eigen- spaces in HW4. Diagonalize T if it is diagonalizable. Justify your claims and show your work. You can freely use any conclusions from the solutions of HW4. Uploaded filesMATH 225 WINTER 2024 HOMEWORK 4 Question 4. (5 PTS) Let T: Mat. .(R)- Mat. (R) be T(A) = AT. We know from Example 5 of Feb. 6 lecture that T is a linear transformation from Mat. .(R) to itself. Calculate the eigenvalues and their associated eigenspaces. Hint: Follow the definitions. Finding the matrix representation is not necessary. The answers may depend on the value of s. Solution. Eigenvalues. By definition AER is an cigruvalue if and only if A-T(A) =AA (31) for some nonzero matrix A. Now for this particular matrix A we have T(AT) = (AT) = A while at the same time T()-T(AA)=AT(A)-XA (32) Therefore (X -1) 4=0. As A is a nonzero matrix there must hold A = 1. Consequently A= 1 are the only candidates for eigenvalues. o Am 1 in an eigenvalue since T(!) = / where I is the identity matrix. o Am -1 is an eigemalue when n > 1 since the matrix A with ajg = =daj = 1 and every other By -0 satisfies T(A) =-4. Therefore when n = 1, the only eigenvalue is 1, when n > 1 the eigenvalues are $1. Eigenspaces. For n = 1 the cigemspace associated to A= 1 is Span(( 1 )) = Mata,:(R). In the following we consider n > 1, or equivalently n22 o A=l. We have T - {AE Mat.-(R)|AT - A) the subspace of symmetric matrices. o Am-1. We have T_1= [AE Mate,-(R)| AT =-4) (33) the subspace of anti-symmetric matrices