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Chapter 10: Problem 60

Find all possible rational forms for a \(6 \times 6\) matrix over \(\mathbf{R}\) with minimal polynomial: (a) \(\quad m(t)=\left(t^{2}-2 t+3\right)(t+1)^{2}\) (b) \(\quad m(t)=(t-2)^{3}\)

Short Answer

Expert verified
For the minimal polynomial \(m(t)=(t^2-2t+3)(t+1)^2\), the possible rational forms are: 1. One 2x2 Jordan block with eigenvalue -1, one 4x4 Jordan block with eigenvalue -1, and two 2x2 Jordan blocks associated with the quadratic term (\(t^2-2t+3\)). 2. Two 3x3 Jordan blocks with eigenvalue -1, and two 2x2 Jordan blocks associated with the quadratic term (\(t^2-2t+3\)). For the minimal polynomial \(m(t)=(t-2)^3\), the only possible rational form is: 1. One 4x4 Jordan block with eigenvalue 2, and one 2x2 Jordan block with eigenvalue 2.

Step by step solution

01

(a) Determine the eigenvalues and their multiplicities

In case (a), the minimal polynomial is \(m(t) = (t^2 - 2t + 3)(t + 1)^2\). We can see that the eigenvalues are: - \(t = -1\), with multiplicity 2 (from the factor \((t + 1)^2\)). - Complex eigenvalues from the quadratic factor \(t^2 - 2t + 3\)
02

(a) Determine the number of Jordan blocks

Now we need to determine the number of Jordan blocks for each eigenvalue. The number of Jordan blocks for an eigenvalue equals its multiplicity in the minimal polynomial. In our case, we have: - Eigenvalue of \(-1\) has 2 Jordan blocks. - Quadratic term (\(t^2 - 2t + 3\)) corresponds to a single block.
03

(a) Determine the possible sizes of Jordan blocks

Now we determine the possible sizes of the Jordan blocks for each eigenvalue. For eigenvalue -1: - There are 2 Jordan blocks. Since we have a 6x6 matrix, the sum of the sizes of the Jordan blocks must be equal to 6. So, we can have either a 2x2 Jordan block and a 4x4 Jordan block or two 3x3 Jordan blocks. For the complex eigenvalues (\(t^2 - 2t + 3\) term): - The block associated with the quadratic term will be of size 2x2, so there will be two such blocks.
04

(a) Forming all possible rational forms

Now we can form all the possible Jordan forms based on the above analysis for case (a): Option 1: - One 2x2 Jordan block with eigenvalue -1 - One 4x4 Jordan block with eigenvalue -1 - Two 2x2 Jordan blocks associated with the quadratic term (\(t^2 - 2t + 3\)) Option 2: - Two 3x3 Jordan blocks with eigenvalue -1 - Two 2x2 Jordan blocks associated with the quadratic term (\(t^2 - 2t + 3\))
05

(b) Determine the eigenvalues and their multiplicities

In case (b), the minimal polynomial is \(m(t) = (t - 2)^3\). We can see that the only eigenvalue is \(t = 2\), with multiplicity 3.
06

(b) Determine the number of Jordan blocks

The number of Jordan blocks for the eigenvalue \(t = 2\) equals its multiplicity in the minimal polynomial, which is 3 in our case.
07

(b) Determine the possible sizes of Jordan blocks

Since there are 3 Jordan blocks for eigenvalue 2 and we have a 6x6 matrix, the sum of the sizes of the Jordan blocks must be equal to 6. The possible Jordan block sizes are: - One 4x4 block and one 2x2 block
08

(b) Forming all possible rational forms

Now we can form all the possible Jordan forms based on the above analysis for case (b): Option 1: - One 4x4 Jordan block with eigenvalue 2 - One 2x2 Jordan block with eigenvalue 2

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Minimal Polynomial
The minimal polynomial of a matrix provides a compact way to encapsulate the algebraic properties of that matrix. It is the monic polynomial of least degree such that when the matrix is substituted into the polynomial, the result is the zero matrix. One can find the minimal polynomial by taking factors corresponding to the matrix's eigenvalues and their geometric multiplicities. This polynomial is crucial for determining the matrix's rational canonical form or Jordan form because it reveals the simplest algebraic constraints the matrix satisfies.

For instance, in the given exercise, we have two different minimal polynomials for a 6x6 matrix over \( \mathbf{R} \). For part (a), the minimal polynomial suggests there are eigenvalues at \( t = -1 \) and complex roots from \( t^2 - 2t + 3 \), implying a direct connection to the potential Jordan blocks in the matrix's structure. In this case, the minimal polynomial hints at the presence of both real and complex eigenvalues, which shapes the matrix's rational forms.
Jordan Blocks
Jordan blocks play a vital role in the study of linear algebra, as they are the building blocks of the Jordan canonical form鈥攁 form of a matrix that exhibits its eigenvalues on the diagonal and potential ones immediately above the diagonal in each block. Each Jordan block corresponds to a single eigenvalue and can have a size ranging from 1x1 (a single entry) to nxn (where n is the size of the entire matrix, if the algebraic multiplicity allows).

In the exercise, we consider the number and sizes of Jordan blocks that align with eigenvalues -1 and 2 for the respective minimal polynomials provided. Particularly, for eigenvalue -1, we identify two Jordan blocks and then discuss how these blocks could be organized within a 6x6 matrix. The possible configurations of these blocks are key in then constructing all the possible rational forms of the matrix.
Eigenvalues
Eigenvalues are one of the most fundamental concepts in linear algebra. They are special numbers associated with a matrix that provide insight into the matrix's properties, such as whether it is invertible and its behavior under certain transformations. The roots of the characteristic polynomial of a matrix are its eigenvalues which inform us about the matrix's invariant directions under the associated linear transformation.

In the discussed exercise, determining the eigenvalues is the first step in the process of finding rational forms of a matrix. We found that eigenvalues could be real or complex, and their multiplicities play a crucial role in deciding the structure and number of Jordan blocks that compose the matrix's rational form. For example, an eigenvalue of \( t = -1 \) with multiplicity 2 signals that there will be two blocks associated with this eigenvalue in the Jordan form.

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Most popular questions from this chapter

Determine the invariant subspace of \(A=\left[\begin{array}{ll}2 & -4 \\ 5 & -2\end{array}\right]\) viewed as a linear operator on (a) \(\mathbf{R}^{2},(\mathrm{b}) \mathbf{C}^{2}\).

Let \(T: V\) be linear. Suppose, for \(v \in V, T^{k}(v)=0\) but \(T^{k-1}(v) \neq 0 .\) Prove (a) The set \(S=\left\\{v, T(v), \ldots, T^{k-1}(v)\right\\}\) is linearly independent (b) The subspace \(W\) generated by \(S\) is \(T\) -invariant (c) The restriction \(\hat{T}\) of \(T\) to \(W\) is nilpotent of index \(k\) (d) Relative to the basis \(\left\\{T^{k-1}(v), \ldots, T(v), v\right\\}\) of \(W\), the matrix of \(T\) is the \(k\) -square Jordan nilpotent block \(N_{k}\) of index \(k\) (see Example 10.5 ). (a) Suppose \\[ a v+a_{1} T(v)+a_{2} T^{2}(v)+\cdots+a_{k-1} T^{k-1}(v)=0 \\] Applying \(T^{k-1}\) to \((*)\) and using \(T^{k}(v)=0,\) we obtain \(a T^{k-1}(v)=0 ;\) because \(T^{k-1}(v) \neq 0, a=0\) Now applying \(T^{k-2}\) to \((*)\) and using \(T^{k}(v)=0\) and \(a=0,\) we find \(a_{1} T^{k-1}(v)=0 ;\) hence, \(a_{1}=0\) Next applying \(T^{k-3}\) to \((*)\) and using \(T^{k}(v)=0\) and \(a=a_{1}=0,\) we obtain \(a_{2} T^{k-1}(v)=0 ;\) hence \(a_{2}=0 .\) Continuing this process, we find that all the \(a\) 's are \(0 ;\) hence, \(S\) is independent. (b) Let \(v \in W\). Then \\[ v=b v+b_{1} T(v)+b_{2} T^{2}(v)+\cdots+b_{k-1} T^{k-1}(v) \\] Using \(T^{k}(v)=0,\) we have \\[ T(v)=b T(v)+b_{1} T^{2}(v)+\cdots+b_{k-2} T^{k-1}(v) \in W \\] Thus, \(W\) is \(T\) -invariant. (c) By hypothesis, \(T^{k}(v)=0 .\) Hence, for \(i=0, \ldots, k-1\) \\[ \hat{T}^{k}\left(T^{i}(v)\right)=T^{k+i}(v)=0 \\] That is, applying \(\hat{T}^{k}\) to each generator of \(W\), we obtain \(0 ;\) hence, \(\hat{T}^{k}=\mathbf{0}\) and so \(\hat{T}\) is nilpotent of index at most \(k .\) On the other hand, \(\hat{T}^{k-1}(v)=T^{k-1}(v) \neq 0 ;\) hence, \(T\) is nilpotent of index exactly \(k\) (d) For the basis \(\left\\{T^{k-1}(v), T^{k-2}(v), \ldots, T(v), v\right\\}\) of \(W\) \\[ \begin{array}{l} \hat{T}\left(T^{k-1}(v)\right)=T^{k}(v)=0 \\ \hat{T}\left(T^{k-2}(v)\right)=T^{k-1}(v) \\ \hat{T}\left(T^{k-3}(v)\right)=T^{k-2}(v) \end{array} \\] $$\hat{T}(T(v)) \quad=\quad T^{2}(v)$$ $$\hat{T}(v) \quad=\quad T(v)$$ Hence, as required, the matrix of \(T\) in this basis is the \(k\) -square Jordan nilpotent block \(N_{k}\)

Prove that \(Z(u, T)=Z(v, T)\) if and only if \(g(T)(u)=v\) where \(g(t)\) is relatively prime to the \(T\) -annihilator of \(u\).

Suppose \(V=W_{1} \oplus \ldots \oplus W_{r} .\) Let \(E_{i}\) denote the projection of \(V\) into \(W_{i} .\) Prove (i) \(E_{i} E_{j}=0, i \neq j\) (ii) \(I=E_{1}+\cdots+E_{r}\)

Prove Theorem 10.3: Suppose \(W\) is \(T\) -invariant. Then \(T\) has a triangular block representation \(\left[\begin{array}{ll}A & B \\ 0 & C\end{array}\right],\) where \(A\) is the matrix representation of the restriction \(\hat{T}\) of \(T\) to \(W\) We choose a basis \(\left\\{w_{1}, \ldots, w_{r}\right\\}\) of \(W\) and extend it to a basis \(\left\\{w_{1}, \ldots, w_{r}, v_{1}, \ldots, v_{s}\right\\}\) of \(V\). We have \\[ \begin{array}{l} \hat{T}\left(w_{1}\right)=T\left(w_{1}\right)=a_{11} w_{1}+\cdots+a_{1 r} w_{r} \\ \hat{T}\left(w_{2}\right)=T\left(w_{2}\right)=a_{21} w_{1}+\cdots+a_{2 r} w_{r} \\ \begin{array}{l} \hat{T}\left(w_{r}\right)=T\left(w_{r}\right)=a_{r 1} w_{1}+\cdots+a_{n} w_{r} \\\ T\left(v_{1}\right)=b_{11} w_{1}+\cdots+b_{1 r} w_{r}+c_{11} v_{1}+\cdots+c_{1 s} v_{s} \\ T\left(v_{2}\right)=b_{21} w_{1}+\cdots+b_{2 r} w_{r}+c_{21} v_{1}+\cdots+c_{2 s} v_{s} \\ T\left(v_{s}\right)=b_{s 1} w_{1}+\cdots+b_{s r} w_{r}+c_{s 1} v_{1}+\cdots+c_{s s} v_{s} \end{array} \end{array} \\] \\[ T\left(v_{s}\right)=b_{s 1} w_{1}+\cdots+b_{s r} w_{r}+c_{s 1} v_{1}+\cdots+c_{s s} v_{s} \\] But the matrix of \(T\) in this basis is the transpose of the matrix of coefficients in the above system of equations (Section 6.2). Therefore, it has the form \(\left[\begin{array}{ll}A & B \\ 0 & C\end{array}\right],\) where \(A\) is the transpose of the matrix of coefficients for the obvious subsystem. By the same argument, \(A\) is the matrix of \(\hat{T}\) relative to the basis \(\left\\{w_{i}\right\\}\) of \(W\)
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