Section 29.4 Concepts
In this section.
Subsection 29.4.1 Scalar-triangular form
A matrix in scalar-triangular form is reasonably simple: its rank, determinant, characteristic polynomial, and eigenvalues are immediately evident, and it is halfway to being row reduced. The name of the form is significant beyond just describing the shape of the entries. A scalar-triangular matrix can be decomposed into a scalar matrix and a βpurelyβ triangular matrix, both additively and multiplicatively. First, an additive example:Subsection 29.4.2 Generalized eigenvectors
Suppose \lambda is an eigenvalue of an n \times n matrix A\text{,} and \uvec{x} is a corresponding eigenvector in \R^n\text{.} By definition, this means thatSubsection 29.4.3 The similarity pattern of scalar-triangular form
Rather than work in the abstract, let's go back over Discovery 29.2, in which we were considering a matrix A in the same similarity class as a scalar-triangular matrixSubsection 29.4.4 Scalar-triangularization procedure
We'll now use the pattern of the example from Discovery 29.2 analyzed in the previous subsection to create a scalar-triangularization procedure.Procedure 29.4.1. To scalar-triangularize an n \times n matrix A\text{,} if possible.
- Compute the characteristic polynomial c_A(\lambda) of A by computing \det (\lambda I - A)\text{,} then determine the eigenvalues of A by solving the characteristic equation c_A(\lambda) = 0\text{.} If A has more than one eigenvalue, stop β A cannot be put into scalar-triangular form. If A has one and only one eigenvalue, write \lambda_0 for this eigenvalue, and continue.
- Determine a basis for the eigenspace E_{\lambda_0}(A) by solving the homogeneous linear system (\lambda_0 I - A) \uvec{x} = \zerovec\text{.} If the basis in this step has n vectors in it, go to the last step. Otherwise, continue.
- Extend the basis for E_{\lambda_0}(A) computed in the previous step to a basis for the generalized eigensubspace of degree 2\text{,} E_{\lambda_0}^2(A)\text{.} Do this by solving the homogeneous linear system (\lambda_0 I - A)^2 \uvec{x} = \zerovec\text{,} and using the already obtained basis for E_{\lambda_0}(A) = E_{\lambda_0}^1(A) as the first part of the solution. If the basis in this step has n vectors in it, go to the last step. Otherwise, continue.
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Continue in this fashion, extending to a basis of E_{\lambda_0}^3(A) (i.e. the null space of (\lambda_0 I - A)^3), and then to a basis for E_{\lambda_0}^4(A) (i.e. the null space (\lambda_0 I - A)^4), and so on, until you reach a point where your basis has n vectors.
- Once your collection of independent generalized eigenvectors has grown to n vectors, place these vectors, in the order you obtained them (i.e. first the vectors from E_{\lambda_0}^1(A)\text{,} then the vectors from E_{\lambda_0}^2(A)\text{,} etc.), as the columns of P\text{.}
Remark 29.4.2.
- Procedure 29.4.1 is actually a little more prescriptive than it needs to be. To ensure a scalar-triangular form, all that is really required is that the first column of P come from E_{\lambda_0}^1(A)\text{,} the second column come from E_{\lambda_0}^2(A)\text{,} and so on. (And still requiring linear independence of the final set of columns.) But since each generalized eigensubspace is contained in the next, taking two vectors from one generalized eigensubspace could be viewed as taking one from the current eigensubspace and one from the next. So in practice we might as well take as many new linearly independent generalized eigenvectors as possible from each new null space computation.
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Once again, it is not necessary to compute \inv{P} to determine the block-diagonal form matrix \inv{P} A P\text{.} One could use row reduction to compute \inv{P} A P\text{,} as in Subsection 26.4.2. But also one could go back to the pattern of similarity from Subsection 26.3.2.
We know that the eigenvalue \lambda_0 will appear repeated down the diagonal of the form matrix. So the \nth[j] column of \inv{P}AP will look like
\begin{equation*} \begin{bmatrix} c_1 \\ \vdots \\ c_{j-1} \\ \lambda_0 \\ 0 \\ \vdots \\ 0 \end{bmatrix} \text{,} \end{equation*}where the c_i are unknown entries above the diagonal that we'd like to determine. From our general similarity pattern, we must have
\begin{equation*} A\uvec{p}_j = c_1 \uvec{p}_1 + \dotsb + c_{j-1} \uvec{p}_{j-1} + \lambda_0 \uvec{p}_k \text{.} \end{equation*}Rearranging this a little differently than usual, we get
\begin{equation*} (A - \lambda_0 I) \uvec{p}_j = c_1 \uvec{p}_1 + \dotsb + c_{j-1} \uvec{p}_{j-1} \text{.} \end{equation*}Therefore, the entries above the diagonal in the \nth[j] column of a scalar-triangular form matrix \inv{P}AP are precisely the coefficients needed to express (A - \lambda_0 I) \uvec{p}_j as a linear combination of the previous columns \uvec{p}_1,\dotsc,\uvec{p}_{j-1}. These coefficients can be determined by row reducing
\begin{equation*} \left[\begin{array}{cccc|c} \uvec{p}_1 \amp \uvec{p}_2 \amp \cdots \amp \uvec{p}_{j-1} \amp (A - \lambda_0 I) \uvec{p}_j \end{array}\right] \text{.} \end{equation*}Though if you're going to do this much row reducting, it might easier to just reduce
\begin{equation*} \left[\begin{array}{c|c} P \amp AP \end{array}\right] \quad\rowredarrow\quad \left[\begin{array}{c|c} I \amp \inv{P}AP \end{array}\right]\text{.} \end{equation*}