![]() ![]() See eigsvdgui in Numerical Computing with MATLAB or Cleve's Laboratory. The resulting diagonal contains the singular values. Now a two-sided QR iteration reduces the off diagonal to negligible size. Use a Householder operating from the left to zero a column and then another Householder operating from the right to zero most of a row. Make our test matrix rectangular by inserting a couple of rows of the identity matrix. The limit is the diagonal containing the eigenvalues. Now the QR iteration works on just two vectors, the diagonal and the off-diagonal. (The computation is done on half of the matrix, but we show the entire array.)īy symmetry the six Householders that zero the columns also zero the rows. The eigenvalues of a bump are a complex conjugate pair of eigenvalues of the input matrix. This example has one bump in rows 3 and 4. So the final matrix may have 2-by-2 bumps on the diagonal. The remaining subdiagonals require just one or two iterations each.Īll this is done with real arithmetic, although a real, nonsymmetric matrix may have complex eigenvalues. The next two rows require three iterations each. The element below the diagonal in the last row is the initial target it requires four iterations. The iteration count is shown in the title. The corresponding diagonal element is an eigenvalue. Now the QR algorithm gradually reduces most subdiagonal elements to roundoff level, so they can be set to zero. The corresponding values of v that satisfy the equation are the right eigenvectors. The values of that satisfy the equation are the eigenvalues. The result is known as a Hessenberg matrix (don't let spell-checkers change that to Heisenberg matrix.) The eigenvalue problem is to determine the solution to the equation Av v, where A is an n -by- n matrix, v is a column vector of length n, and is a scalar. The initial reduction uses n-2 Householder similarites to introduce zeroes below the subdiagonal a column at a time. Here is a static picture of the starting matrix. ![]() The starting matrix for all three variants is based on flipping the Rosser matrix.
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