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:
“Once a child has demonstrated his capacity for independent functioning in any area, his lapses into dependent behavior, even though temporary, make the mother feel that she is being taken advantage of....What only yesterday was a description of the child’s stage in life has become an indictment, a judgment.”
—Elaine Heffner (20th century)
“A sound mind in a sound body, is a short, but full description of a happy state in this World: he that has these two, has little more to wish for; and he that wants either of them, will be little the better for anything else.”
—John Locke (1632–1704)
“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)