What condition must a non-zero vector \(\mathbf v\) satisfy to be an eigenvector of \(A\)?

Why is \(\mathbf0\) not called an eigenvector?

Check whether \((1,0)\) is an eigenvector of \(A=\begin{pmatrix}5&2\\0&3\end{pmatrix}\).

If \(\mathbf v\) is an eigenvector with eigenvalue \(4\), what is \(A(3\mathbf v)\)?

Find an eigenvector for \(A=\begin{pmatrix}2&0\\0&5\end{pmatrix}\) with eigenvalue \(2\).

Find an eigenvector for \(A=\begin{pmatrix}1&0\\0&-3\end{pmatrix}\) with eigenvalue \(-3\).

Check whether \((1,1)\) is an eigenvector of \(A=\begin{pmatrix}2&1\\1&2\end{pmatrix}\).

Check whether \((1,2)\) is an eigenvector of \(A=\begin{pmatrix}1&0\\0&3\end{pmatrix}\).

Find eigenvectors for \(A=\begin{pmatrix}2&1\\1&2\end{pmatrix}\) with eigenvalue \(1\).

Find eigenvectors for \(A=\begin{pmatrix}4&1\\0&4\end{pmatrix}\) with eigenvalue \(4\).

Explain why any non-zero scalar multiple of an eigenvector is also an eigenvector.

Interpret an eigenvector of a stretching matrix physically.

For which \(k\) is \((1,1)\) an eigenvector of \(A=\begin{pmatrix}2&k\\1&3\end{pmatrix}\)?

For which \(a\) is \((1,-1)\) an eigenvector of \(A=\begin{pmatrix}a&2\\3&1\end{pmatrix}\)?

Find eigenvectors for \(A=\begin{pmatrix}3&2\\0&1\end{pmatrix}\) for each eigenvalue.

Find all \(t\) such that \((1,t)\) is an eigenvector of \(A=\begin{pmatrix}0&1\\2&1\end{pmatrix}\).

A matrix has eigenvectors \((1,0)\) and \((1,1)\) with eigenvalues \(2\) and \(5\). Find \(A(3,2)\).

A student finds \((0,0)\) while solving \((A-\lambda I)\mathbf v=\mathbf0\) and lists it as an eigenvector. Explain the error.

Prove that eigenvectors belonging to distinct eigenvalues of a \(2\times2\) matrix are linearly independent.

Explain why solving \(A\mathbf v=\mathbf0\) usually does not find eigenvectors for a non-zero eigenvalue.