DLA Scalar-triangular form

Section 35.3 Scalar-triangular form

What.

Upper triangular with repeated \(\lambda\) down the diagonal:

\begin{equation*} \inv{P}AP = \begin{bmatrix} \lambda\\ \amp \lambda \amp \amp \ast \\ \amp \amp \ddots \\ \amp 0 \amp \amp \lambda \\ \amp \amp \amp \amp \lambda \end{bmatrix}\text{.} \end{equation*}

When.

Matrix \(A\) has a single eigenvalue \(\lambda\) with algebraic multiplicity \(n\text{.}\)

How.

Compute a basis for the eigenspace \(E_\lambda(A)\) (which is the null space of \(\lambda I - A\)), and use these as the first columns of \(P\text{.}\) If this has not produced enough columns for \(P\text{,}\) compute a basis for the next generalized eigensubspace \(E^2_\lambda(A)\) (which is the null space of \((\lambda I - A)^2\)). Choose as many new vectors as possible from this basis that remain linearly independent when lumped together with the already chosen basis vectors from \(E_\lambda(A)\text{.}\) Use these new vectors as the next columns of \(P\text{.}\) If this still has not produced enough columns for \(P\text{,}\) continue by adding vectors from a basis for \(E^3_\lambda(A)\text{,}\) then from \(E^4_\lambda(A)\text{,}\) and so on.

Result.

The entries along the diagonal are the eigenvalue \(\lambda\text{.}\) The entries above the diagonal in the \(\nth[j]\) column of \(\inv{P} A P\) are precisely the scalars required to express \((A - \lambda I)\uvec{p}_j\) as a linear combination of \(\uvec{p}_1,\dotsc,\uvec{p}_{j-1}\text{.}\)