How does QR algorithm work?

How does QR algorithm work?

The QR algorithm was developed in the late 1950s by John G. F. Francis and by Vera N. Kublanovskaya, working independently. The basic idea is to perform a QR decomposition, writing the matrix as a product of an orthogonal matrix and an upper triangular matrix, multiply the factors in the reverse order, and iterate.

How do you use QR decomposition to find eigenvalues?

2 Answers

1. Let X by a symmetric matrix.
2. Let X1=X, and iterate the following:
3. Given Xk, write a QR decomposition Xk=QkRk, and let Xk+1=RkQk;
4. The matrices sequence Xn converges to some diagonal matrix D with the eigenvalues on the diagonal; you retrieve the corresponding eigenvectors as the columns of ∏iQi.

What happens if you apply the unshifted QR algorithm to an orthogonal matrix?

When you factor an orthogonal matrix Q in the form QR, R is just the identity. Thus the unshifted QR algorithm makes no progress toward getting this matrix to upper triangular (or, if symmetric, diagonal) form.

How is the QR method used to find eigenvalues?

The algorithm which produces this sequence of matrices uses the QR factorization of a matrix; almost every mathematical software package uses a variation of this strategy to compute the eigenvalues of a matrix. To introduce the algorithm, we \frst establish some properties that underlie this QR method for \fnding eigenvalues.

How does the QR algorithm apply to linear algebra?

It follows that the columns of Q are the eigenvectors (which are the principal objects that are computed!) and that R has the same eigenvalues as A. size 2 × 2, in which case R i i has a pair of complex conjugate eigenvalues of A (such as 2 + i and 2 − i ).

How does the unshift QR algorithm compute the diagonal matrix?

Assume further that A = QΛQT is the eigenvalue decomposition of A, where QT = LU has an LU decomposition and the diagonal entries of U are nonnegative. Then, the unshift QR algorithm on A computes iterates A1, A2, A3, …, that converge to a diagonal matrix.

Which is matrix has the same eigenvalues as AK?

It turns out that the sequence A0, A1, …, have the same eigenvalues and for any large integer K the matrix AK is usually close to being upper-triangular. Since the eigenvalues of an upper-triangular matrix lie on its diagonal, the iteration above will allow us to read off the eigenvalues of A from the diagonal entries of AK.