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Appendix A The Singular Value Decomposition and the Pseudoinverse 451 These r orthonormal q's can be extended, by Gram-Schmidt, to a complete orthonormal basis q1,. . ., qm. Those are the columns of Qv The i,j entry of the product Q]AQ2 is qjAxj (row times matrix times column), and we know those entries: qJAXj = 0 if; > r (because Axt = 0) q}AXj = qjvjqj (10) if./ < r (because Axf = (JjCjj). But qjqj â 0 except when / = j . Thus the only nonzeros in the product Z = Q\AQ2 are the first r diagonal entries, which are ax,. .., ar. Since QJ = Q1 we have A = Q^Ql as required. We mention again how these vectors x and q fit the four fundamental subspaces of A. The first r x's are an orthonormal basis for the row space, and the last n â r are an orthonormal basis for the nullspace (where Axt = 0). The first r q's are an orthonormal basis for the column space, and the last m â r are an orthonormal basis for the left nullspace. Of course it is no surprise that those bases exist. We know that! What is special about the x's and q\ is that Ax] = afly The matrix takes basis vectors into basis vectors, when the bases are right. By the rules for constructing a matrix, where you apply A to the basis in R" and express the result using the basis in Rm, the matrix with respect to these bases is diagonal. It is Z. EXERCISES A.1 Find the singular value decomposition and the pseudoinverse 0 + of the m by n zero matrix. A.2 Find the singular value decomposition and the pseudoinverse of A = [l l I lj and B= 0 l I 0 I) 0 and C' = l l 0 I) A.3 If an m by n matrix Q has orthonormal columns, what is its pseudoinverse Q + l A.4 (a) Compute A^A and its eigenvalues and eigenvectors and positive definite square root S if 10 A = ⢠10 (b) 6 0 8 Compute the polar decomposition A = QS. Knowing Q, find the reverse form A = S'Q. A.5 What is the minimum length least squares solution v ' = A ' b to Ax "l l l 0 0" 0 0 l l "c" "()' D = 2 2 E
