Eigenvalues and eigenvectors – university Year 1 maths worksheet with answers to download as PDF

Let \(A=\) \(\begin{pmatrix}4&1\\2&3\end{pmatrix}\).

1. Write down and solve the characteristic equation \(\det(A-\lambda I)=0\). [3 marks]
2. Find an eigenvector for each eigenvalue. [4 marks]
3. Check that the trace equals the sum of the eigenvalues and the determinant equals their product. [2 marks]

Find the eigenvalues of each matrix.

1. \(\begin{pmatrix}1&2\\0&3\end{pmatrix}\) [2 marks]
2. \(\begin{pmatrix}2&0&0\\0&3&4\\0&4&9\end{pmatrix}\) [4 marks]
3. A \(2\times2\) matrix has trace 7 and determinant 10. [3 marks]

Find the eigenvalues of each real matrix and give a complex eigenvector for one of them.

1. \(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\) (a rotation through \(90^\circ\)) [3 marks]
2. \(\begin{pmatrix}1&-2\\2&1\end{pmatrix}\) [4 marks]

Let \(A=\) \(\begin{pmatrix}2&1\\1&2\end{pmatrix}\).

1. Find the eigenvalues and eigenvectors of \(A\). [4 marks]
2. Write \(A=PDP^{-1}\) with \(D\) diagonal. [3 marks]
3. Use it to compute \(A^5\). [3 marks]

Let \(N=\) \(\begin{pmatrix}2&1\\0&2\end{pmatrix}\).

1. Find the eigenvalue and its algebraic multiplicity. [2 marks]
2. Find all eigenvectors. What is the geometric multiplicity? [3 marks]
3. Explain why \(N\) cannot be diagonalised. [2 marks]

Let \(A\) have eigenvalues \(2\) and \(5\) (for example \(A=\) \(\begin{pmatrix}4&1\\2&3\end{pmatrix}\)).

1. Find the eigenvalues of \(A^2\), \(A^{-1}\) and \(A+3I\). [4 marks]
2. Find \(\det(A^2)\) and \(\operatorname{tr}(A^2)\) from the eigenvalues, and check with direct calculation. [4 marks]

Each month 10% of customers of Network A switch to Network B, and 20% of customers of Network B switch to Network A. Let \(\mathbf v_n=(a_n,b_n)^T\) be the proportions after \(n\) months, so \(\mathbf v_{n+1}=P\mathbf v_n\) with \(P=\) \(\begin{pmatrix}0.9&0.2\\0.1&0.8\end{pmatrix}\).

1. Show that 1 is an eigenvalue and find the second eigenvalue.
2. Find the eigenvectors.
3. Find the long-term market shares.
4. Starting from \(\mathbf v_0=(1,0)^T\), find \(\mathbf v_n\) and the share of Network A after 5 months.

A rabbit population has juveniles \(J\) and adults \(A\). Each period every adult produces 2 juveniles, 50% of juveniles become adults, and 80% of adults survive. So \(\begin{pmatrix}J'\\A'\end{pmatrix}\)\(=\) \(\begin{pmatrix}0&2\\0.5&0.8\end{pmatrix}\)\(\begin{pmatrix}J\\A\end{pmatrix}\).

1. Write the characteristic equation and find the eigenvalues to 3 decimal places.
2. Which eigenvalue governs the long-term growth? What is the growth rate per period?
3. Find the long-term ratio of adults to juveniles.
4. Explain why the other eigenvalue does not matter in the long run.

Two carriages of mass \(m\) are joined by springs of stiffness \(k\). For displacements \(\mathbf x=(x_1,x_2)\) the equations of motion are \(\ddot{\mathbf x}=-\dfrac km K\mathbf x\) where \(K=\) \(\begin{pmatrix}2&-1\\-1&2\end{pmatrix}\). A normal mode \(\mathbf x=\mathbf v\cos\omega t\) requires \(K\mathbf v=\mu\mathbf v\) with \(\omega^2=\dfrac{k\mu}m\).

1. Find the eigenvalues \(\mu\) of \(K\).
2. Find the eigenvectors and describe the two modes of motion.
3. Find the two angular frequencies if \(k=300\) N/m and \(m=3\) kg.
4. Which mode is faster? Explain physically.

A photo-editing program stretches an image by the matrix \(S=\) \(\begin{pmatrix}3&1\\1&3\end{pmatrix}\) applied to coordinates relative to the centre.

1. Find the eigenvalues of \(S\).
2. Find the directions that are only stretched, not turned.
3. By what factor is each of these directions stretched?
4. Find the area scale factor and check that it equals the product of the eigenvalues.

Two connected rooms have temperature deviations \(\mathbf x=(x_1,x_2)\) from the outside temperature (in \(^\circ\)C). They satisfy \(\dot{\mathbf x}=A\mathbf x\) with \(A=\) \(\begin{pmatrix}-3&1\\1&-3\end{pmatrix}\) (per hour). Initially \(\mathbf x(0)=(4,0)\).

1. Find the eigenvalues and eigenvectors of \(A\).
2. Write the general solution \(\mathbf x(t)=c_1e^{\lambda_1t}\mathbf v_1+c_2e^{\lambda_2t}\mathbf v_2\).
3. Find the constants \(c_1,c_2\) and the temperatures after 1 hour.
4. Explain why both rooms eventually reach the outside temperature.

A tiny website has three pages. Page A links to B and C, page B links to C, and page C links to A. The link matrix is \(L=\) \(\begin{pmatrix}0&0&1\\1/2&0&0\\1/2&1&0\end{pmatrix}\), where each column shows how a page shares its importance among the pages it links to. The ranking vector satisfies \(L\mathbf r=\mathbf r\).

1. Check that each column of \(L\) sums to 1.
2. Solve \(L\mathbf r=\mathbf r\) with \(r_A+r_B+r_C=1\).
3. Which pages rank highest?
4. Explain why page B ranks lower than C although both receive a link from A.

Spiral counts in sunflowers are often consecutive Fibonacci numbers. The Fibonacci matrix is \(Q=\) \(\begin{pmatrix}1&1\\1&0\end{pmatrix}\), and \(Q^n=\) \(\begin{pmatrix}F_{n+1}&F_n\\F_n&F_{n-1}\end{pmatrix}\).

1. Find the eigenvalues of \(Q\). They are \(\varphi=\dfrac{1+\sqrt5}2\) and \(\psi=\dfrac{1-\sqrt5}2\).
2. Find eigenvectors in terms of \(\varphi\) and \(\psi\).
3. Use the diagonalisation to explain Binet's formula \(F_n=\dfrac{\varphi^n-\psi^n}{\sqrt5}\).
4. Evaluate \(\dfrac{\varphi^{10}}{\sqrt5}\) and compare with \(F_{10}=55\).

A machine is either working (W) or broken (B) at the start of each day. A working machine breaks with probability 0.05. A broken machine is repaired by the next day with probability 0.4. The transition matrix is \(T=\) \(\begin{pmatrix}0.95&0.4\\0.05&0.6\end{pmatrix}\).

1. Find the eigenvalues of \(T\).
2. Find the long-term fraction of days on which the machine works.
3. Find the machine state probabilities after 2 days if it works today.
4. How fast does the system approach its long-term behaviour?

A culture of bacteria is counted each generation. The counts (in millions) satisfy \(a_{n+2}=5a_{n+1}-6a_n\) with \(a_0=0\) and \(a_1=1\).

1. Check the entries in the table.
2. Write the recurrence as \(\begin{pmatrix}a_{n+1}\\a_n\end{pmatrix}\)\(=A\)\(\begin{pmatrix}a_n\\a_{n-1}\end{pmatrix}\) and find \(A\).
3. Find the eigenvalues of \(A\) and write \(a_n=\alpha2^n+\beta3^n\). Find \(\alpha,\beta\).
4. Find \(a_{10}\) and the long-run ratio \(a_{n+1}/a_n\).

The stress energy in a metal plate is modelled by the quadratic form \(Q(x,y)=5x^2+4xy+2y^2\), which can be written \(Q=\mathbf x^TS\mathbf x\) with \(S=\) \(\begin{pmatrix}5&2\\2&2\end{pmatrix}\). Engineers look for the principal axes, along which the pattern is simplest.

1. Find the eigenvalues of \(S\).
2. Find unit eigenvectors and check that they are perpendicular.
3. Write \(Q\) in the form \(\lambda_1u^2+\lambda_2v^2\) in the rotated coordinates \((u,v)\).
4. Describe the curve \(Q=6\) and give the lengths of its semi-axes.

Let \(A=\) \(\begin{pmatrix}2&1\\1&3\end{pmatrix}\). The table shows the top-left entry of \(A^n\).

1. Check the entries for \(n=2\) and \(n=3\) by multiplying matrices.
2. Find the characteristic equation and the eigenvalues of \(A\).
3. Use Cayley-Hamilton to show that \(A^2=5A-5I\) and then that \(A^{-1}=\dfrac15(5I-A)\).
4. Explain why the ratio of successive entries tends to the dominant eigenvalue and compute it.

Let \(A_k=\) \(\begin{pmatrix}k&1\\1&k\end{pmatrix}\), where \(k\) is a real constant.

1. Show that the eigenvalues are \(k+1\) and \(k-1\) for every \(k\), and find the eigenvectors.
2. For which \(k\) is \(A_k\) invertible?
3. For which \(k\) is the system \(\dot{\mathbf x}=A_k\mathbf x\) stable, so that all solutions tend to zero?
4. For \(k=0\) describe \(A_0\) geometrically.

Engineers describe four prototype control systems \(\dot{\mathbf x}=A\mathbf x\) by the trace and determinant of \(A\). For a \(2\times2\) matrix, \(\lambda=\dfrac{\tau\pm\sqrt{\tau^2-4\Delta}}2\) with \(\tau=\operatorname{tr}A\) and \(\Delta=\det A\).

1. Find the eigenvalues of each system.
2. Classify each as stable node, stable spiral, saddle or unstable node.
3. Which systems tend to zero for every starting point?
4. For P2, write the form of the solutions and the oscillation frequency.

Let \(\mathbf v_1,\mathbf v_2\) be eigenvectors of a matrix \(A\) for distinct eigenvalues \(\lambda_1\ne\lambda_2\). Prove that they are linearly independent.

1. Suppose \(\alpha\mathbf v_1+\beta\mathbf v_2=\mathbf 0\). Apply \(A\) to get a second equation.
2. Eliminate \(\mathbf v_2\) and deduce \(\alpha=0\).
3. Deduce \(\beta=0\).
4. Show that \(A\) and \(A^T\) have the same eigenvalues, and that for a \(2\times2\) matrix \(\lambda_1+\lambda_2=\operatorname{tr}A\).