7.5. Exercises

Verify that for each of the following matrices, the given vector $v$ is an eigenvector, and find the corresponding eigenvalue.

1. (i)

2. (ii)

3. (iii)

4. (iv)

5. (v)

6. (vi)

   .

Find the characteristic equation and eigenvalues for each of the following linear transformations or matrices:

1. (i)

   $T\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -x\hfill \\ \hfill 2y\hfill \end{array}\right)$

2. (ii)

   $T\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -4x-3y\hfill \\ \hfill 3x+2y\hfill \end{array}\right)$

3. (iii)

   $T\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \\ \hfill z\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -3z+x\hfill \\ \hfill z+2y\hfill \\ \hfill -3z\hfill \end{array}\right)$

4. (iv)

   .

5. (v)

   $A$ is the $2\times 2$ matrix of reflection about the line $y=x$.

6. (vi)

7. (vii)

8. (viii)

In each of the cases from Exercise 7.5.2, and for each of the eigenvalues found, determine the corresponding eigenspaces, and state their dimensions.

Is the subset $\{\left(\begin{array}{c}\hfill t+1\hfill \\ \hfill 2t+2\hfill \\ \hfill 0\hfill \end{array}\right)|t\in ℝ\}\subset {ℝ}^3$ a subspace? If so, verify all the axioms. If not, explain which axiom it doesn’t satisfy.

Is the subset $\{\left(\begin{array}{c}\hfill t\hfill \\ \hfill 2t\hfill \\ \hfill t^2\hfill \end{array}\right)|t\in ℝ\}\subset {ℝ}^3$ a subspace? If so, verify all the axioms. If not, explain which axiom it doesn’t satisfy.

If $T:{ℝ}^2\to {ℝ}^2$ is a linear transformation, and $W\subset {ℝ}^2$ is a subspace, then prove that $TW:=\{Tw|w\in W\}$ is also a subspace of ${ℝ}^2$.

Let $T$ be the linear transformation corresponding to the matrix . Is there a 1-dimensional subspace $W\subset {ℝ}^2$ such that $TW$ is not a 1-dimensional subspace? Justify your answer.

Let be the matrix of an invertible linear transformation. Proceeding as in the proof of Theorem 7.3.3, show that $A$ has:

1. (i)

   two distinct real eigenvalues if and only if ${(a-d)}^2-4b^2>0$,

2. (ii)

   exactly one real eigenvalue if and only if ${(a-d)}^2-4b^2=0$, and

3. (iii)

   no real eigenvalue if and only if .

A rotation matrix $R_{\theta }$ is of the above form. Use the above to find all values of $\theta$ for which $R_{\theta }$ has real eigenvalues. Compare your answer to Example 7.1.

This exercise consists of diagonalizing a symmetric matrix, which will be covered more fully in MATH220. Let with $b\ne 0$. By Theorem 7.3.3 we have two distinct real eigenvalues ${\lambda }_1,{\lambda }_2$; let $v_i=\left(\begin{array}{c}\hfill x_i\hfill \\ \hfill y_i\hfill \end{array}\right)$ be an eigenvector for ${\lambda }_i$ for $i=1,2$. Assume $v_1$ and $v_2$ are each of length 1.

1. (i)

   By Theorem 7.3.3 we know that $v_1$ and $v_2$ are perpendicular to each other. Write this conclusion as an equation in terms of $x_1,x_2,y_1,$ and $y_2$.

2. (ii)

   Let . Prove that $P^{t}P=P{P}^t=I_2$.

3. (iii)

   Prove that

4. (iv)

   Use the above to prove that

In this sense, we have used the matrix $P$ to diagonalize $A$. From the diagonal form it is easy to see what the eigenvalues are.