DLA Triangular-block form

Processing math: 100%

Skip to main content

$\require{cancel}\DeclareMathOperator{\RREF}{RREF} \DeclareMathOperator{\adj}{adj} \DeclareMathOperator{\proj}{proj} \DeclareMathOperator{\matrixring}{M} \DeclareMathOperator{\poly}{P} \DeclareMathOperator{\Span}{Span} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\nullity}{nullity} \DeclareMathOperator{\null}{null} \DeclareMathOperator{\uppermatring}{U} \DeclareMathOperator{\trace}{trace} \DeclareMathOperator{\dist}{dist} \DeclareMathOperator{\neg}{neg} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\im}{im} \newcommand{\basisfont}[1]{\mathcal{#1}} \newcommand{\bbrac}[1]{\bigl(#1\bigr)} \newcommand{\Bbrac}[1]{\Bigl(#1\Bigr)} \newcommand{\irst}[1][1]{{#1}^{\mathrm{st}}} \newcommand{\ond}[1][2]{{#1}^{\mathrm{nd}}} \newcommand{\ird}[1][3]{{#1}^{\mathrm{rd}}} \newcommand{\nth}[1][n]{{#1}^{\mathrm{th}}} \newcommand{\leftrightlinesubstitute}{\scriptstyle \overline{\phantom{xxx}}} \newcommand{\inv}[2][1]{{#2}^{-{#1}}} \newcommand{\abs}[1]{\left\lvert #1 \right\rvert} \newcommand{\degree}[1]{{#1}^\circ} \newcommand{\iddots}{{\kern3mu\raise1mu{.}\kern3mu\raise6mu{.}\kern3mu \raise12mu{.}}} \newcommand{\blank}{-} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\ci}{\mathrm{i}} \newcommand{\cconj}[1]{\bar{#1}} \newcommand{\lcconj}[1]{\overline{#1}} \newcommand{\cmodulus}[1]{\left\lvert #1 \right\rvert} \newcommand{\iso}{\simeq} \newcommand{\abctriangle}[1]{\triangle #1} \newcommand{\mtrxvbar}{\mathord{|}} \newcommand{\utrans}[1]{{#1}^{\mathrm{T}}} \newcommand{\rowredarrow}{\xrightarrow[\text{reduce}]{\text{row}}} \newcommand{\bidentmatfour}{\begin{bmatrix} 1 \amp 0 \amp 0 \amp 0 \\ 0 \amp 1 \amp 0 \amp 0 \\ 0 \amp 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 0 \amp 1\end{bmatrix}} \newcommand{\uvec}[1]{\mathbf{#1}} \newcommand{\zerovec}{\uvec{0}} \newcommand{\bvec}[2]{#1\,\uvec{#2}} \newcommand{\ivec}[1]{\bvec{#1}{i}} \newcommand{\jvec}[1]{\bvec{#1}{j}} \newcommand{\kvec}[1]{\bvec{#1}{k}} \newcommand{\injkvec}[3]{\ivec{#1} - \jvec{#2} + \kvec{#3}} \newcommand{\abray}[1]{\overrightarrow{#1}} \newcommand{\norm}[1]{\left\lVert #1 \right\rVert} \newcommand{\unorm}[1]{\norm{\uvec{#1}}} \newcommand{\bigcdot}{\mathbin{\large\boldsymbol{\cdot}}} \newcommand{\dotprod}[2]{#1\bigcdot#2} \newcommand{\udotprod}[2]{\dotprod{\uvec{#1}}{\uvec{#2}}} \newcommand{\crossprod}[2]{#1\times#2} \newcommand{\ucrossprod}[2]{\crossprod{\uvec{#1}}{\uvec{#2}}} \newcommand{\uproj}[2]{\proj_{\uvec{#2}} \uvec{#1}} \newcommand{\adjoint}[1]{{#1}^\ast} \newcommand{\matrixOfplain}[2]{{\left[#1\right]}_{#2}} \newcommand{\rmatrixOfplain}[2]{{\left(#1\right)}_{#2}} \newcommand{\rmatrixOf}[2]{\rmatrixOfplain{#1}{\basisfont{#2}}} \newcommand{\matrixOf}[2]{\matrixOfplain{#1}{\basisfont{#2}}} \newcommand{\invmatrixOfplain}[2]{\inv{\left[#1\right]}_{#2}} \newcommand{\invrmatrixOfplain}[2]{\inv{\left(#1\right)}_{#2}} \newcommand{\invmatrixOf}[2]{\invmatrixOfplain{#1}{\basisfont{#2}}} \newcommand{\invrmatrixOf}[2]{\invrmatrixOfplain{#1}{\basisfont{#2}}} \newcommand{\stdmatrixOf}[1]{\left[#1\right]} \newcommand{\ucobmtrx}[2]{P_{\basisfont{#1} \to \basisfont{#2}}} \newcommand{\uinvcobmtrx}[2]{\inv{P}_{\basisfont{#1} \to \basisfont{#2}}} \newcommand{\uadjcobmtrx}[2]{\adjoint{P}_{\basisfont{#1} \to \basisfont{#2}}} \newcommand{\coordmapplain}[1]{C_{#1}} \newcommand{\coordmap}[1]{\coordmapplain{\basisfont{#1}}} \newcommand{\invcoordmapplain}[1]{\inv{C}_{#1}} \newcommand{\invcoordmap}[1]{\invcoordmapplain{\basisfont{#1}}} \newcommand{\similar}{\sim} \newcommand{\inprod}[2]{\left\langle\, #1,\, #2 \,\right\rangle} \newcommand{\uvecinprod}[2]{\inprod{\uvec{#1}}{\uvec{#2}}} \newcommand{\orthogcmp}[1]{{#1}^{\perp}} \newcommand{\vecdual}[1]{{#1}^\ast} \newcommand{\vecddual}[1]{{#1}^{\ast\ast}} \newcommand{\dd}[2]{\frac{d{#1}}{d#2}} \newcommand{\ddx}[1][x]{\dd{}{#1}} \newcommand{\ddt}[1][t]{\dd{}{#1}} \newcommand{\dydx}{\dd{y}{x}} \newcommand{\dxdt}{\dd{x}{t}} \newcommand{\dydt}{\dd{y}{t}} \newcommand{\intspace}{\;} \newcommand{\integral}[4]{\int^{#2}_{#1} #3 \intspace d{#4}} \newcommand{\funcdef}[3]{#1\colon #2\to #3} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&}$

Section 35.4 Triangular-block form

What.

A block-diagonal form matrix

$\begin{equation*} \inv{P}AP = \begin{bmatrix} U_1 \\ \amp U_2 \\ \amp \amp \ddots \\ \amp \amp \amp U_\ell \end{bmatrix}\text{,} \end{equation*}$

where each block

$U$ corresponds to a specific eigenvalue

$\lambda$ of

$A\text{,}$ has size equal to the algebraic multiplicity of

$\lambda\text{,}$ and is in scalar-triangular form with

$\lambda$ down the diagonal.

When.

The characteristic polynomial of

$A$ factors completely as

$\begin{equation*} c_A(\lambda) = (\lambda - \lambda_1)^{m_1} (\lambda - \lambda_2)^{m_2} \dotsm (\lambda - \lambda_\ell)^{m_\ell}\text{.} \end{equation*}$

How.

Follow the scalar-triangular form procedure (Procedure 29.4.1) with

$\lambda = \lambda_1\text{,}$ but stop when you have

$m_1$ linearly independent vectors. Use these vectors as the first columns of

$P\text{.}$ Repeat with

$\lambda = \lambda_2$ to get the next

$m_2$ columns of

$P\text{.}$ And so on.

Result.

Each block

$U_j$ will be as in the described result for Scalar-triangular form.