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Exercises 15 10. Let A = λ1 0 0 λ2 and B = λ1 r 0 λ2 , where λ1 = λ2. Find a triangular matrix of the form S = 1 x 0 1 such that S−1AS = B. Hint: Work with the equation AS = SB instead of S−1AS = B. 11. Let A = λ 1 0 λ and B = λ r 0 λ . Assuming r = 0, find a diagonal matrix S such that S−1AS = B. 12. Recall that the trace of a square matrix is the sum of the diagonal entries. Show that if A is m × n and B is n × m, then AB and BA have the same trace. 13. The Hadamard product of two m × n matrices R and S is the m × n matrix with rijsij in entry (i, j) thus, it is entrywise multiplication. We denote the Hadamard product by R ◦ S. Show that if A is m × n and B is n × m, then the trace of AB is the sum of the entries of A ◦ BT (equivalently, the sum of the entries of AT ◦ B). (If you did Exercise 12 in the usual way, then this should follow easily.) 14. Show that similar matrices have the same trace. Hint: Use Exercise 12. 15. Let A be an n × n complex matrix. Show that the trace of A∗A is n ∑ i=1 ∑n j=1 |aij|2. 16. Find a pair of similar matrices A and B such that A∗A and B∗B have different traces. Hint: Use Exercises 10 and 15. 17. A complex matrix U is said to be unitary if U −1 = U ∗ that is, if UU ∗ = U ∗ U = I. Show that if A is an m × n matrix and B = U ∗ AV , where U and V are unitary matrices of sizes m × m and n × n, respectively, then A∗A and B∗B have the same trace. 18. A linear transformation T : V → V is called a projection if T 2 = T . For example, the matrix P = Ik ⊕ 0n−k is a projection matrix. In this exercise, you will show that every projection on a finite-dimensional vector space can be represented by a matrix of this form. Let T be a projection on a vector space V. Let N = {x ∈ V | T x = 0} and R = {x ∈ V | T x = x}. (a) Show that N and R are subspaces of V. (b) Show that for any x ∈ V, the vector x − T x is in N and the vector T (x) is in R. (c) Use part (b) to show that V = R + N . Furthermore, since R ∩ N = 0, we have V = R ⊕ N . (d) Now let V be finite dimensional. Let n = dim(V) and k = dim(R). Let {r1,..., rk} be a basis for R, and let {rk+1,..., rn} be a basis for N . Then B = {r1,..., rn} is a basis for V. Show that the matrix for T with respect to this basis is Ik ⊕ 0n−k.