The $3 \times 3$ identity matrix
Terms
| Term | This example | Why |
|---|---|---|
| array rank | $2$ | |
| shape | $(3, 3)$ | |
| array size | $3 \cdot 3 = 9$ | |
| matrix rank | $\operatorname{rank} I = 3$ | The $3 \times 3$ identity matrix has rank $3$, because its columns are linearly independent. |
| positive definite | ✓ | |
| positive semidefinite | ✓ |
Distinctions using this example
- Positive definite vs. positive semidefinite: The $3 \times 3$ identity matrix is positive definite, because $x^{\top} I x = \lVert x \rVert^2 > 0$ for every nonzero $x$. So it is also positive semidefinite.