 Solution

For an efficient QR algorithm, first reduce a general real matrix to upper Hessenberg matrix by orthogonal similarities. For a real symmetric matrix, use Householder tridiagonalization; the resulting symmetric tridiagonal matrix is the usual input in this setting. These structures are preserved by suitable QR algorithm implementations and greatly reduce the work per iteration.

The output is the eigenvalues, together with an orthogonal reduction when the factors are accumulated. In the symmetric case the target is a diagonal matrix, whose diagonal entries are the real eigenvalues; the accumulated orthogonal columns are eigenvectors. For a general real matrix, the target is the real form in a Schur decomposition: an upper block triangular matrix with diagonal blocks of size one or two. The two-by-two blocks represent complex conjugate eigenvalue pairs. An arbitrary unshifted iteration need not converge, so these describe the intended converged output, with appropriate shifts and block deflation in the QR algorithm.