Finding eigenvalues and eigenspaces for the matrix A

eigenvalues-eigenvectorslinear algebramatrices

A is a matrix that reflects vectors x $\in$ $\mathbb{R}^3$ about a fixed plane P in $\mathbb{R}^3$. Find all eigenvalues and eigenspaces of A. Does A admit an eigenbasis for $\mathbb{R}^3$? Why or why not?

Another part of this question: Let a = $\begin{bmatrix}\alpha & \beta & \gamma \end{bmatrix}^T$ $\in$ $\mathbb{R}^3$ be nonzero and fixed. Consider the linear transformation T : $\mathbb{R}^3 \to \mathbb{R}^3$ given by T(x) = a $\times$ x. Find all real eigenvalues of T and all real eigenspaces of T. Does T admit an eigenbasis for $\mathbb{R}^3$? Why or why not?

for the first part I don't know how I would find an eigenvalue for a matrix I don't know, I would assume A would look like a reflection matrix but I've looked online and I can't find anything that tells me what a 3$\times$3 reflection matrix would look like.

for the second part, I know that a would look like $\begin{bmatrix}\alpha\\\ \beta \\\ \gamma \end{bmatrix}$, but wouldn't a $\times$ x look like $\begin{bmatrix}\beta x_3 – \gamma x_2 & \gamma x_1 – \alpha x_3 & \alpha x_2 – \beta x_1 \end{bmatrix}$? And if so how would you even find the eigenvalues and eigenspaces of that?

Any help is appreciated, thanks!

Best Answer

The key to doing this problem efficiently is to understand the geometric interpretation of an eigenvalue. Recall that for a linear transformation $T:\Bbb R^n \to \Bbb R^n$, we say that $x \in \Bbb R^n$ is an eigenvector of $T$ associated with the eigenvalue $\lambda$ if it is a non-zero vector for which $T(x) = \lambda x$.

For the first problem, our linear transformation is $T_A(x) = Ax$. It is given that $T$ takes a vector and reflects it across the plane $P$. Suppose that $v_1$ is a non-zero vector perpendicular to $P$.

  • I claim that $v_1$ must be an eigenvector of $T_A$. Do you see why? What is the eigenvalue associated with $v_1$?
  • Take any two linearly independent vectors $v_2,v_3$ that are orthogonal to $v_1$ (that is, two vectors from the plane $P$). I claim that these are also eigenvectors. Do you see why? What are the associated eigenvalues?
  • $v_1,v_2,v_3$ form a basis for $\Bbb R^3$, and each vector is an eigenvector. So, $A$ admits an eigenbasis for $\Bbb R^3$.

For the second problem, our linear transformation is $T(x) = \alpha \times x$.

  • By the properties of the cross product, $T(x)$ must be orthogonal to $x$. There is exactly one real number $\lambda$ for which it is possible to have $x \neq 0$ and $T(x) = \lambda x$, what is this $\lambda$? If $x$ is an eigenvector associated with this $\lambda$, what must be true about $x$?
  • From the above, we have found out that $T$ has only one real eigenvalue, and the eigenspace associated with that eigenvalue is $1$-dimensional. Since we have no more eigenvectors with which to construct an eigenbasis, we conclude that $T$ does not admit an eigenbasis for $\Bbb R^3$.

As I explained above, using the specific entries for the matrices associated with these transformations turns out not to be a "nice" approach. If you are interested in seeing what these matrices might look like, then you should know that the reflection matrix from the first part is the Householer transformation $A = I - 2vv^T$, and the cross-product matrix for the second part is $[\alpha]_{\times}$, as is explained in this section of the cross-product wiki page.

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