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:
“The next Augustan age will dawn on the other side of the Atlantic. There will, perhaps, be a Thucydides at Boston, a Xenophon at New York, and, in time, a Virgil at Mexico, and a Newton at Peru. At last, some curious traveller from Lima will visit England and give a description of the ruins of St. Paul’s, like the editions of Balbec and Palmyra.”
—Horace Walpole (1717–1797)
“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)
“An intentional object is given by a word or a phrase which gives a description under which.”
—Gertrude Elizabeth Margaret Anscombe (b. 1919)