We have recently been looking at how we can use a special case of Francis's QR transformation to reduce a real symmetric matrix

The columns of the matrix of transformations

and, since the product of

which is known as the spectral decomposition of

Last time we saw how we could efficiently apply the Householder transformations in-place, replacing the elements of

**M**to a diagonal matrix**Λ**by first applying Householder transformations to put it in tridiagonal form and then using shifted Givens rotations to zero out the off diagonal elements.The columns of the matrix of transformations

**V**and the elements on the leading diagonal of**Λ**are the unit eigenvectors and eigenvalues of**M**respectively and they consequently satisfy**M**×

**V**=

**V**×

**Λ**

and, since the product of

**V**and its transpose is the identity matrix**M**=

**V**×

**Λ**×

**V**

^{T}

which is known as the spectral decomposition of

**M**.Last time we saw how we could efficiently apply the Householder transformations in-place, replacing the elements of

**M**with those of the matrix of accumulated transformations**Q**and creating a pair of arrays to represent the leading and off diagonal elements of the tridiagonal matrix. This time we shall see how we can similarly improve the implementation of the Givens rotations.Full text...