Example · Undergraduate
Terms
| Term | This example | Why |
|---|
| entrywise exponential |
\[\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\]
|
The entrywise exponential of the $2 \times 2$ zero matrix is the matrix of ones, because $e^0 = 1$ for each entry.
|
| matrix exponential |
\[e^{0} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\]
|
The matrix exponential of the $2 \times 2$ zero matrix is the identity matrix, because every term of $I + 0 + 0^2 / 2! + \cdots$ after $I$ is zero.
|
| matrix rank |
$\operatorname{rank} 0 = 0$
|
The $2 \times 2$ zero matrix has rank $0$, because its column space is $\setdef{0}$.
|
| nullity |
$\operatorname{nullity} 0 = 2$
|
The $2 \times 2$ zero matrix has nullity $2$, because it sends every vector in $\reals^2$ to $0$.
|
| singular matrix |
✓ |
The $2 \times 2$ zero matrix is singular, because its determinant is $0$.
|
| defective matrix |
✗ |
The $2 \times 2$ zero matrix is not defective, because its eigenvalue $0$ has multiplicity $2$ and every nonzero vector is an eigenvector, so it has $2$ linearly independent eigenvectors.
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