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Section 35.2 Block-diagonal form

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What.
\begin{equation*} \inv{P} A P = \begin{bmatrix} B_1 \\ \amp B_2 \\ \amp \amp \ddots \\ \amp \amp \amp B_\ell \end{bmatrix}\text{,} \end{equation*}

where each B_j is a square block.

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When.

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There exists a complete set of independent subspaces of \R^n (or \C^n\text{,} as appropriate), with each subspace in the collection also A-invariant.

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How.

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Compute a basis for each subspace in the set of independent, A-invariant spaces. Use all of these basis vectors together as the columns of P\text{,} with vectors from each subspace basis grouped together.

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Result.

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Each block corresponds to one of the invariant subspaces, so the number of blocks is the same as the number of invariant subspaces. Furthermore, each block will have size equal to the dimension of the corresponding A-invariant subspace. If W = \Span\{\uvec{w}_1,\dotsc,\uvec{w}_k\} is the A-invariant subspace corresponding to one of the blocks of the form matrix B=\inv{P}AP\text{,} then the \nth[j] column of that block is precisely the set of coefficients required to express A\uvec{w}_j as a linear combination of the basis vectors \{\uvec{w}_1,\dotsc,\uvec{w}_k\}\text{.}