December 2019 Archives

Spryer Francis

Last time we saw how we could use a sequence of Householder transformations to reduce a symmetric real matrix M to a symmetric tridiagonal matrix, having zeros everywhere other than upon the leading, upper and lower diagonals, which we could then further reduce to a diagonal matrix Λ using a sequence of Givens rotations to iteratively transform the elements upon the upper and lower diagonals to zero so that the columns of the accumulated transformations V were the unit eigenvectors of M and the elements on the leading diagonal of the result were their associated eigenvalues, satisfying

    M × V = V × Λ

and, since the transpose of V is its own inverse

    M = V × Λ × VT

which is known as the spectral decomposition of M.
Unfortunately, the way that we used Givens rotations to diagonalise tridiagonal symmetric matrices wasn't particularly efficient and I concluded by stating that it could be significantly improved with a relatively minor change. In this post we shall see what it is and why it works.

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