Eigenvalues and Orthogonality: math practice, College – download the PDF

1 Is it an eigenvector? ★★★

Let \(A=\begin{pmatrix}1&2\\3&0\end{pmatrix}\). For each vector, decide whether it is an eigenvector of \(A\). If so, give its eigenvalue.

1. \(\mathbf{v}=(1,1)\)
2. \(\mathbf{w}=(1,2)\)

2 Dot product and length ★★★

Let \(\mathbf{u}=(2,-1,3)\) and \(\mathbf{w}=(4,5,-1)\).

1. Compute \(\mathbf{u}\cdot\mathbf{w}\). Are the vectors orthogonal?
2. Compute \(\|\mathbf{u}\|\).

3 A first characteristic polynomial ★★★

Find the eigenvalues of \(A=\begin{pmatrix}5&2\\1&4\end{pmatrix}\).

4 Triangular matrix ★★★

Without expanding a determinant, find the eigenvalues of \(T=\begin{pmatrix}2&7&1\\0&-3&5\\0&0&4\end{pmatrix}\), and verify them with the trace.

5 Angle between two vectors ★★★

Find the angle between \(\mathbf{u}=(1,0,1)\) and \(\mathbf{w}=(0,1,1)\).

6 A quick projection ★★★

Project \(\mathbf{b}=(6,2)\) onto the line spanned by \(\mathbf{a}=(2,1)\).

7 True or false: powers and shifts ★★★

Suppose \(A\mathbf{v}=4\mathbf{v}\) for a nonzero vector \(\mathbf{v}\). Compute \(A^2\mathbf{v}\), \(A^3\mathbf{v}\) and \((A+3I)\mathbf{v}\) in terms of \(\mathbf{v}\). Then decide: is it true that if \(0\) is an eigenvalue of \(A\), then \(A\) is not invertible?

8 Finding eigenvectors ★★★

Find the eigenvalues and one eigenvector for each, for \(A=\begin{pmatrix}4&1\\2&3\end{pmatrix}\).

9 Diagonalize and take a power ★★★

Let \(A=\begin{pmatrix}5&2\\2&2\end{pmatrix}\).

1. Find its eigenvalues and eigenvectors.
2. Write \(A=PDP^{-1}\).
3. Use the eigenvalues to explain why \(A^3\) has eigenvalues \(216\) and \(1\), and compute \(A^3\).

10 A 3 by 3 characteristic polynomial ★★★

Let \(M=\begin{pmatrix}1&2&0\\2&1&0\\0&0&4\end{pmatrix}\). Find the characteristic polynomial, the eigenvalues, and an eigenvector for each eigenvalue.

11 A rotation has no real eigenvalues ★★★

Let \(R=\begin{pmatrix}0&-1\\1&0\end{pmatrix}\), a quarter-turn counterclockwise in the plane. Show that \(R\) has no real eigenvalue, and explain this geometrically.

12 Gram-Schmidt in the plane ★★★

Apply Gram-Schmidt to \(\mathbf{v}_1=(1,2)\) and \(\mathbf{v}_2=(4,1)\), then give an orthonormal basis.

13 Distance from a point to a line ★★★

A drone at position \((1,7)\) wants to know how far it is from the straight path through the origin in the direction \((1,2)\) (units: hundreds of feet). Find the closest point on the path and the distance.

14 A three-point fit ★★★

A coffee shop records the number of iced drinks sold (in dozens) at temperatures \(-1,0,1\) (in tens of degrees from a reference value): \((-1,1),(0,2),(1,6)\). Use the normal equations to find the least squares line \(y=a+bx\).

15 Diagonalizable or not? ★★★

Decide whether each matrix is diagonalizable and justify your answer.

1. \(A=\begin{pmatrix}3&1\\0&3\end{pmatrix}\)
2. \(B=\begin{pmatrix}3&0\\0&3\end{pmatrix}\)

16 Finding a missing entry ★★★

The matrix \(A=\begin{pmatrix}1&k\\2&4\end{pmatrix}\) has \(5\) as an eigenvalue. Find \(k\) and the other eigenvalue.

17 Orthogonal diagonalization of a symmetric matrix ★★★

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

1. Show that \((1,1,1)\) is an eigenvector and find its eigenvalue.
2. Show that \((1,-1,0)\) and \((1,1,-2)\) are eigenvectors with eigenvalue \(1\), and that the three vectors are pairwise orthogonal.
3. Find the trace of \(S^5\).

18 Gram-Schmidt in three dimensions ★★★

Apply Gram-Schmidt to \(\mathbf{v}_1=(1,1,0)\), \(\mathbf{v}_2=(1,0,1)\), \(\mathbf{v}_3=(0,1,1)\).

19 Least squares for plant growth ★★★

A botanist measures a seedling’s height (in inches) at weeks \(0,1,2,3\): \(1,\,2,\,2,\,4\).

1. Find the least squares line \(y=a+bx\).
2. Compute the sum of squared residuals.
3. Predict the height at week \(5\).

20 Eigenvectors of symmetric matrices ★★★

Let \(A\) be a symmetric matrix, with \(A\mathbf{u}=\lambda\mathbf{u}\), \(A\mathbf{w}=\mu\mathbf{w}\) and \(\lambda\neq\mu\). Prove that \(\mathbf{u}\cdot\mathbf{w}=0\).

21 A projection matrix ★★★

Let \(\mathbf{a}=(1,2,2)\) and \(P=\dfrac{\mathbf{a}\mathbf{a}^T}{\mathbf{a}\cdot\mathbf{a}}\).

1. Write \(P\) and compute \(P\mathbf{b}\) for \(\mathbf{b}=(3,0,3)\).
2. Find the eigenvalues of \(P\).

22 Two apps, long-term behavior ★★★

Each month, \(10\%\) of the users of app A switch to app B, and \(20\%\) of the users of app B switch to app A. Let \(\mathbf{x}_n=(a_n,b_n)\) be the number of users (in thousands) after \(n\) months, so \(\mathbf{x}_{n+1}=M\mathbf{x}_n\) with \(M=\begin{pmatrix}0.9&0.2\\0.1&0.8\end{pmatrix}\). Initially \(\mathbf{x}_0=(150,150)\).

1. Find the eigenvalues and eigenvectors of \(M\).
2. Write \(\mathbf{x}_0\) in terms of the eigenvectors and give a formula for \(\mathbf{x}_n\).
3. What happens as \(n\) grows?