Last time we saw how we could use a sequence of Householder transformations to reduce a symmetric real matrix

and, since the transpose of

which is known as the spectral decomposition of

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.

**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**×

**V**

^{T}

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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