Characteristic vector of zero matrix

Question

Can we talk about characteristic vector of a zero matrix? If yes, What is the characteristic vector of the zero matrix?

Answer 1

Any basis of $\mathbb{R}^n$ (orthonormal ones are preferable) happen to be a basis of eigenvectors for the $n \times n$ zero matrix. For example, for the zero matrix \begin{bmatrix} 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0 \end{bmatrix} any of the following matrices (whose columns make a basis of $\mathbb{R}^4$) $$\begin{bmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{bmatrix} \quad\quad \begin{bmatrix} 1&-1&-1&-1\\ 1&1&-1&-1\\ 1&0&2&-1\\ 1&0&0&3 \end{bmatrix} \quad\quad \frac{1}{\sqrt{2}}\begin{bmatrix} 1&-1&-1&-1\\ 1&1&0&0\\ 1&0&1&0\\ 1&0&0&1 \end{bmatrix}$$ will work.

Answer 2

A “characteristic vector”, or “eigenvector”, $v$ corresponding to a given eigenvalue $\lambda$ of matrix $A$ is a vector such that $Av= \lambda v$. Since the 0 matrix times any vector is the 0 vector, the only eigenvalue is 0 and every non-zero vector is a characteristic vector.