๐ท Main Diagonal
In a square matrix $A$, the main diagonal consists of entries where row index equals column index: $a_{ii}$.
- This diagonal forms the backbone for defining matrix types and operations.
๐งฎ Matrix Types That Depend on the Diagonal
| Matrix Type | Diagonal Role |
|---|---|
| Identity Matrix | All diagonal entries are 1, others are 0 |
| Diagonal Matrix | Only diagonal entries may be nonzero |
| Upper Triangular | All entries below the diagonal are 0 |
| Lower Triangular | All entries above the diagonal are 0 |
| Symmetric Matrix | $a_{ij} = a_{ji}$, diagonal anchors symmetry |
๐ Determinant Calculation
- For triangular matrices, the determinant is the product of diagonal entries.
- In Gaussian elimination, pivot positions lie on the diagonal and determine solvability.
๐ง Eigenvalue Theory
- In diagonalizable matrices, the diagonal often holds the eigenvalues.
- In Jordan canonical form, diagonal entries reflect algebraic multiplicities.
๐ข Trace of a Matrix
- The trace is the sum of diagonal entries: $$ \text{tr}(A) = \sum_{i=1}^{n} a_{ii} $$
- Invariant under similarity transformations and appears in characteristic polynomials.
Last updated on