Use Sage to do your computations for you.
See Appendix B.4: Scalar-triangular form.
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Each of the following matrices has a single eigenvalue \(\lambda = 5\).
Work through the scalar-triangularization procedure for each of them.
When you have formed the transition matrix \(P\),
compute the form matrix \(P^{-1} A P\) without explicitly computing \(P^{-1}\).
(One way you could do this is go back to the general similarity pattern and compute the coefficients to express each \(A\mathbf{p}_j\) as a linear combination of \(\mathbf{p}_1,\mathbf{p}_2,\dotsc,\mathbf{p}_n\).
Another way is to use the row reduction procedure as in Subsection 26.4.2.)
- \(\begin{bmatrix} 31 & 13 & 78 \\ -4 & 3 &-12 \\ -8 & -4 & -19 \end{bmatrix}\)
- \(\begin{bmatrix} 11 & 2 & 1 \\ -8 & 2 & -1 \\ -16 & -5 & 2 \end{bmatrix}\)
- \(\begin{bmatrix} -16 & 7 & -21 & -14 \\ -9 & 8 & -9 & -6 \\ 18 & -6 & 23 & 12 \\ 0 & 0 & 0 & 5 \end{bmatrix}\)
- \(\begin{bmatrix} 8 & 7 & -3 & 3 \\ -15 & -19 & 10 & -9 \\ -27 & -28 & 16 & -8 \\ 3 & 20 & -9 & 15 \end{bmatrix}\)
- \(\begin{bmatrix} 4 & 3 & 4 & -1 \\ 11 & -24 & -40 & 9 \\ -6 & 16 & 27 & -5 \\ 10 & -26 & -36 & 13 \end{bmatrix}\)
- \(\begin{bmatrix} -2 & 5 & 4 & 4 \\ 9 & -1 & -5 & -5 \\ 9 & -6 & 0 & -5 \\ -32 & 22 & 18 & 23 \end{bmatrix}\)
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Carry out steps a, b, c below for each of the following matrices.
- \(A = \begin{bmatrix} -5 & 3 & 1 \\ -4 & 2 & 1 \\ -4 & 3 & 0 \end{bmatrix}\)
- \(A = \begin{bmatrix} -3 & 6 & 3 & 2 \\ -2 & 3 & 2 & 2 \\ -1 & 3 & 0 & 1 \\ -1 & 1 & 2 & 0 \end{bmatrix}\)
- Calculate and factor the characteristic polynomial \(c_A(\lambda)\). From this factored form, determine the eigenvalues and their algebraic multiplicities.
- Calculate a basis for each eigenspace of \(A\). For each eigenvalue, is its geometric multiplicity equal to or less than its algebraic multiplicity?
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Determine a basis for each generalized eigenspace of \(A\) in the same manner as in the scalar-triangularization procedure.
That is, for each eigenvalue \(\lambda\) of \(A\), extend (if possible) the basis for \(E_\lambda(A)\) you determined in step b to a basis for \(E_\lambda^2(A)\).
Then extend further (if possible) to a basis for \(E_\lambda^3(A)\).
And so on, until you have a basis for some \(E_\lambda^k(A)\) that has \(m_\lambda\) vectors in it, where \(m_\lambda\) is the algebraic multiplicity of \(\lambda\).
Identify the parts of Theorem 29.6.1 that justify that what you have produced is a basis for the generalized eigenspace \(G_\lambda(A)\).
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