Definition: Eigenvalues of a matrix

Question

1) Can a non-square matrix have eigenvalues? Why?

2) True or false: If the characteristic polynomial of a matrix A is p($\lambda$)=$\lambda$^2+1, then A is invertible. Thank you!

Answer 1

For 1) No, it has to be a square matrix by definition.

To see why, consider the following:

Recall that for an eigenvector $v$ and and an eigenvalue $\lambda$, you have that $Av$ = $\lambda v$.

Now suppose that dim(v) = n x 1. That means that dim(Av) = n x 1 and dim($\lambda v$) = n x 1. If A is not square then dim(A) must be m x n where $m \neq n$. But then you have that dim($Av$) = (m x n) * (n x1) = m x 1. But we just said that dim(Av) = n x 1 . Thus contradiction.

In regards to your second question, this post answers it completely.
Is a matrix with characteristic polynomial $t^2 +1$ invertible?

Answer 2

Hint. (1) Try it! For example, can you solve $$\pmatrix{1&2&3\cr4&5&6\cr}\pmatrix{v_1\cr v_2\cr v_3\cr} =\lambda\pmatrix{v_1\cr v_2\cr v_3\cr}\ ?$$

(2) You should know

• the connection between whether or not $A$ is invertible and $\det(A)$;
• the connection between $\det(A)$ and the eigenvalues of $A$;
• the connection between the eigenvalues of $A$ and its characteristic polynomial.