Matrix Analysis Linear Algebra SVD Speech or Presentation

Matrix Analysis Linear Algebra SVD - Speech or Presentation Example It can be easily checked that A, z-A, (z-A)-1 commute and thus are diagonalizable simultaneously. Furthermore, it can be easily be checked directly that if ÃŽ » is an eigenvalue of A with eigenvector V, and (z-ÃŽ »)-1 is an eigenvalue corresponding also to v. Therefore, A, z-A and (z-A)-1 have the same spectral projector PÃŽ » of A= the spectral projector P(z-ÃŽ »)-1of (z-A)-1, and, therefore, the spectral decomposition of (z-A)-1 is thus; 1c.) Given a square matrix M its resolvent is the matrix-valued function 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 Green’s 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 determine 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

Business Economics 2 Essay Example | Topics and Well Written Essays - 1500 words - 3

Business Economics 2 - Essay Example In other words it is a closed private economy. Households supply labor to firms which in turn pay wages to the former. They buy goods and services produced by the firms. Next we introduce the government. The government is both a producer and a consumer at the same time. It is not only actively involved in production and consumption but also in charging taxes from and giving subsidies to the first two groups. In the next stage we introduce financial institutions such as banks. They enable the flows to be made smoother through their services such as cheques, credit cards and so on. However, still this is a domestic economy, though there are financial institutions in the rest of the world as well. So we finally introduce international trade, i.e. imports and exports. This is where the problem of balance of payments comes up. Balance of payments is the sum total of all imports and exports between a particular country (e.g. Britain) and the rest of the world in monetary terms. Therefore global financial institutions also come into the system. The circular flow diagram, indeed, adequately represents the fact that what is paid by a member of a given sector, say, firms to a member of another sector, say, households, is income for the latter while it’s expenditure for the former. However, in itself it’s a static model of a dynamic series of national and international flows. This is where its inadequacy as a representative model of income flows shows up. The balance of payments problem of Britain or for that matter of any other country is a dynamic one which necessitates a dynamic modeling structure to adequately capture the hidden forces of change. For instance in 2006, Britain’s total exports were equal to  £ 369,691 million while imports were equal to  £ 424,128 million. Thus the current account balance recorded a deficit of  £ 54,437 million in 2006 (Annual Blue Book of Statistics,