Consider an n x n matrix A such that A2 = 0, with rank A = r. (In Example 58 we consider
Chapter 7, Problem 61(choose chapter or problem)
Consider an n x n matrix A such that A2 = 0, with rank A = r. (In Example 58 we consider the case when n = 3 and r = 1.) Show that A is similar to the block matrixMatrix B has r blocks of the form J along the diagonal, with all other entries being 0. (Hint: Mimic the approach outlined in Exercise 58. Pick a basis U|,...,Cr of the image if A, write 5/ = Aw, for / = 1,..., r, and expand v\,..., vr to a basis v\,..., vr> U[,..., um of the kernel of A. Show that 5i, w;j, V 2, i>2........5r, wr, U].........um is a basis of IR", and show that B is the matrix of T(x) = Ax with respect to this basis.)
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