Matrix Analysis Linear Algebra SVD Speech or Presentation

Matrix Analysis Linear Algebra SVD - Speech or introduction ExampleIt hindquarters be easily checked that A, z-A, (z-A)-1 commute and thus ar aslantizable simultaneously. Furthermore, it can be easily be checked directly that if is an eigenvalue of A with eigenvector V, and (z-)-1 is an eigenvalue corresponding also to v. Therefore, A, z-A and (z-A)-1 have the same religious projector P of A= the spectral projector P(z-)-1of (z-A)-1, and, therefore, the spectral decomposition of (z-A)-1 is thus1c.) Given a straightforwardly matrix M its resolvent is the matrix-valued dish of a square matrix A its resolvent is the matrix-valued function RA(z)=(zI-A)-1, defined for all z C and I is a n*n identity matrix.In infinite dimensions the resolvent is also called the Greens function. Since the resolvent RA(z)is nothing else but f(A) for f(t)=(z-t)-1=1/z-t its spectral decomposition is exactly what is expected.The diagonals entries i,j of are the singular values of A. The m columns of U and the N columns of V are the left-singular and right-singular vectors of A. One application that uses SVD is the pseudoinverse.A+=V+U*, where + is the pseudoinverse of , which is formed by replacing every non-zero diagonal entry by its reciprocal and getting the transpose of the resulting matrix. It is also possible to use SVD of A to get the orthogonal matrix R closest to the range of A. The closeness of fit is measured by the Frobenius norm of R-A. The solution is the product UV* the orthogonal matrix would have the decomposition UIV* where I is the identity matrix, so that if