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.
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