For problems 1-6 find eigenpairs for the following matrices.
\({\bf A} = \begin{pmatrix}2 & 1 \\ 1 & 2 \end{pmatrix}\)
\({\bf A} = \begin{pmatrix}3 & -2 \\ 3 & -1 \end{pmatrix}\)
\({\bf A} = \begin{pmatrix}3 & -2 \\ 2 & -2 \end{pmatrix}\)
\({\bf A} = \begin{pmatrix}1 & i \\ -i & 1 \end{pmatrix}\)
\({\bf A} = \begin{pmatrix}3 & 2 & 2 \\ 1 & 4 & 1 \\ -2 & -4 & -1 \end{pmatrix}\)
\({\bf A} = \begin{pmatrix}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix}\)
For problems 7 - 9 find all the eigenvectors associated with the given eigenvalue.
\( \ABld = \begin{pmatrix} 1 & 0 & 0\\ -1 & 2 & -1\\ 1 & -1 & 2 \end{pmatrix} \) with eigenvalue \(\lambda = 1\).
\(\ABld = \begin{pmatrix} 2 & -1 & 0\\ 0 & 3 & 0\\ -7 & -7 & -5 \end{pmatrix} \) with eigenvalue \(\lambda =2\).
\(\ABld = \begin{pmatrix} 3 & 7 & 5 & 5\\ 0 & 1 & 0 & 0\\ 0 & -5 & -2 & -5\\ 0 & 2 & 0 & 3 \end{pmatrix} \) with eigenvalue \(\lambda = 3\).
Find the eigenpairs for a matrix of the form \[\ABld = \begin{pmatrix} a & b \\ -b & a \end{pmatrix}\]
Find the eigenpairs for a matrix of the form \[\ABld = \begin{pmatrix} a & b\\ b & -a \end{pmatrix}\]
Consider the following matrix \[\ABld = \begin{pmatrix} 2\alpha & 1\\ -1 & 0 \end{pmatrix}.\] For what values of \(\alpha\) are the eigenvalues real, conjugate pairs, or repeated?
For problems 13-19 find a general (real-valued) solution to the system of differential equations. As \(t \rightarrow \infty\), what happens to the solutions?
\({\bf x}' = \begin{pmatrix} 1 & 1 \\ 4 & -2 \end{pmatrix} {\bf x}\)
\({\bf x}' = \begin{pmatrix} 1 & -1 \\ 1 & 3 \end{pmatrix} {\bf x}\)
\({\bf x}' = \begin{pmatrix} 5 & -1 \\ 3 & 1 \end{pmatrix} {\bf x}\)
\({\bf x}' = \begin{pmatrix} -1 & -4 \\ 1 & -1 \end{pmatrix} {\bf x}\)
\({\bf x}' = \begin{pmatrix} 1 & 2 \\ -5 & -1 \end{pmatrix} {\bf x}\)
\({\bf x}' = \begin{pmatrix} 4 & -2 \\ 8 & -4 \end{pmatrix} {\bf x}\)
\({\bf x}' = \begin{pmatrix}1 & 1 & 2 \\ 1 & 2 & 1 \\ 2 & 1 & 1 \end{pmatrix} {\bf x}\)
For problems 20-23 find a general (real-valued) solution the system of differential equations and construct a fundamental matrix, \(\Psi(t)\).
\({\bf x}' = \begin{pmatrix} 3 & 6 \\ -1 & -2 \end{pmatrix} {\bf x} \)
\({\bf x}' = \begin{pmatrix} 1 & -1 \\ 5 & -3 \end{pmatrix} {\bf x} \)
\({\bf x}' = \begin{pmatrix} 4 & -1 \\ 1 & 2 \end{pmatrix} {\bf x} \)
\({\bf x}' = \begin{pmatrix}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix} {\bf x}\)
For problems 24-26 solve the initial value problem.
\({\bf x}' = \begin{pmatrix} -3 & \sqrt{2} \\ \sqrt{2} & -2 \end{pmatrix} {\bf x}, \hspace{.5 in} {\bf x}(0) = \begin{pmatrix} 2 \\ 1 \end{pmatrix}\)
\({\bf x}' = \begin{pmatrix} -3 & 2 \\ -1 & -1 \end{pmatrix} {\bf x}, \hspace{.5 in} {\bf x}(0) = \begin{pmatrix} 1 \\ -2 \end{pmatrix}\)
\({\bf x}' = \begin{pmatrix} 2 & \frac{3}{2} \\ -\frac{3}{2} & -1 \end{pmatrix} {\bf x}, \hspace{.5 in} {\bf x}(0) = \begin{pmatrix} 3 \\ -2 \end{pmatrix}\)
For problems 27-28 compute \(e^{t\bf A}\) using the fundamental matrix, \(\Psi(t)\).
\({\bf A} = \begin{pmatrix}5 & 3 \\ 5 & 3 \end{pmatrix}\)
\({\bf A} = \begin{pmatrix}1 & -4 \\ 4 & -7 \end{pmatrix}\)
Recall exercise 10 from the chapter on Linear Systems: General Methods and Theory. That exercise considered a coupled system of pendula governed by the second-order system of differential equations \[\begin{aligned} \theta_1^{\prime\prime} & = - \frac{g}{\ell} \theta_1 - \frac{k}{m_1} (\theta_1 - \theta_2) \,, \\ \theta_2^{\prime\prime} & = - \frac{g}{\ell} \theta_2 + \frac{k}{m_2} (\theta_1 - \theta_2) \,. \end{aligned}\] This can be written as a first order linear system \(\xBld^\prime = \ABld \xBld\), where \[\ABld = \begin{pmatrix} 0 & 1 & 0 & 0 \\ - \frac{g}{\ell} - \frac{k}{m_1} & 0 & \frac{k}{m_1} & 0 \\ 0 & 0 & 0 & 1 \\ \frac{k}{m_2} & 0 & -\frac{g}{\ell} - \frac{k}{m_1} & 0 \end{pmatrix} \,.\] Find a general (real-valued) solution to this system.
Show that any \(n\times n\) Hermitian matrix (\(\ABld^* = \ABld\)) has only real eigen-values.
Show that any \(n\times n\) anti-Hermitian matrix (\(\ABld^* = -\ABld\)) has only imaginary eigen-values.
For problems 32-33 show that the following matrices are diagonalizable first. Then, diagonalize them.
\({\bf A} = \begin{pmatrix}2 & 1 \\ 1 & 2 \end{pmatrix}\)
\({\bf A} = \begin{pmatrix}3 & -2 \\ 4 & -1 \end{pmatrix}\)
For which values of \(\alpha\) is the following matrix diagonalizable? \[\ABld = \begin{pmatrix} 1 & \alpha\\ 0 & 1 \end{pmatrix}\]
For problems 35-36 calculate \(e^{t\bf A}\) via diagonalization.
\({\bf A} = \begin{pmatrix}2 & -1 \\ 3 & -2 \end{pmatrix}\)
\({\bf A} = \begin{pmatrix}1 & 5 \\ -1 & 3 \end{pmatrix}\)
Suppose that \(\VBld\) is an invertible \(n\times n\) matrix. Prove the following formula for a given \(n\times n\) matrix \(\ABld\). \[e^{t\VBld\ABld\VBld^{-1}} = \VBld e^{t\ABld} \VBld^{-1}.\] (Hint: Show that both sides satisfy the same initial value problem.)
It is a well know result in linear algebra that any \(2\times 2\) non-diagonalizable matrix can be written as \(\ABld = \VBld\JBld\VBld^{-1}\), where \(\VBld\) is the matrix of generalized eigenvectors associated with the eigenvaue \(a\) and \(\JBld\) is the matrix of the form \[\JBld = \begin{pmatrix} a & 1\\ 0 & a \end{pmatrix}.\] Write an explicit formula for \(e^{t\JBld}\) by computing the \(n^{\mathrm{th}}\) power of \(\JBld\) and using the series representation of \(e^{t\JBld}\). Use this to arrive at a formula for \(e^{t\ABld}\) in terms of \(\VBld\)? Check that this is consistent with the general formula for a \(2\times 2\) matrix with eigen-value \(a\) of multiplicity \(2\). (Hint: use the result from problem 37.)