Description
Let S be a symmetric matrix, and G = G(i,j,θ) be a Givens rotation matrix. Then:
is symmetric and similar to S.
Furthermore, S′ has entries:
where s = sin(θ) and c = cos(θ).
Since G is orthogonal, S and S′ have the same Frobenius norm ||·||F (the square-root sum of squares of all components), however we can choose θ such that S′ij = 0, in which case S′ has a larger sum of squares on the diagonal:
Set this equal to 0, and rearrange:
if
In order to optimize this effect, Sij should be the largest off-diagonal component, called the pivot.
The Jacobi eigenvalue method repeatedly performs rotations until the matrix becomes almost diagonal. Then the elements in the diagonal are approximations of the (real) eigenvalues of S.
Read more about this topic: Jacobi Eigenvalue Algorithm
Famous quotes containing the word description:
“Do not require a description of the countries towards which you sail. The description does not describe them to you, and to- morrow you arrive there, and know them by inhabiting them.”
—Ralph Waldo Emerson (1803–1882)
“The Sage of Toronto ... spent several decades marveling at the numerous freedoms created by a “global village” instantly and effortlessly accessible to all. Villages, unlike towns, have always been ruled by conformism, isolation, petty surveillance, boredom and repetitive malicious gossip about the same families. Which is a precise enough description of the global spectacle’s present vulgarity.”
—Guy Debord (b. 1931)
“Why does philosophy use concepts and why does faith use symbols if both try to express the same ultimate? The answer, of course, is that the relation to the ultimate is not the same in each case. The philosophical relation is in principle a detached description of the basic structure in which the ultimate manifests itself. The relation of faith is in principle an involved expression of concern about the meaning of the ultimate for the faithful.”
—Paul Tillich (1886–1965)