Separation of Subspaces by Volume

Abstract

During the 1995 Christmas break I read the article Angle Between Complementary Subspaces by Ilse Ipsen and Carl Meyer in this MONTHLY [12]. Since I was scheduled to teach a course in matrix analysis the following semester, I made myself some notes, intending to incorporate the ideas of the article into the classroom discussion. But after spending several weeks on such topics as eigenspaces, unitary similarity, vector and matrix norms, and singular values, I found myself short of classroom time. I decided to use the notes to design some take-home questions, and started off with the following problem. (1) Let P be a real, n-by-n idempotent matrix and let U be the image of P and V the image of I, P. Show that R8n = U E V. The students easily handled this problem; but I hoped the exercise reminded them that P is geometrically a projection of Rn onto U along V. I then asked a follow-up question. (2) Assume that n = 2 and that 0 0 P 0 I2. Let 0 be the acute angle between the lines described by the subspaces U and V, and let III P III be the matrix 2-norm of P. Conjecture~ the relationship between 0 and III P 111 A picture of U, V, and 0 was included as part of the question. In class we had previously discussed operator norms and had sketched pictures of unit balls with respect to various vector norms. I hoped the students would recall from a problem in our text [10, p. 312] the meaning of the matrix 2-norm (also called the spectral norm):