05 Eigen Values, Vectors
Eigen Values¶
are the values of \(\lambda\) that satisfy equation
\[ | A - \lambda I | = 0 \]
Properties¶
- Eigen values of upper/lower \(\triangle\)r matrix = diagonal elements
- No of eigen values = order of A
- Sum of eigen values = Sum of diagonal elements
- Product of eigen values = \(|A|\)
- If eigen values of \(A = \lambda\), then
| Matrix | Eigen Value |
|---|---|
| \(A^{-1}\) | \(\frac{1}{\lambda}\) |
| \(A^n\) | \(\lambda^n\) |
| \(A^T\) | \(\lambda\) |
Eigen Vectors¶
are the values of \(X\) that satisfies equation
\[ (A - \lambda I) X = 0 \]
Eigen vector(s) of \(A\) = eigen vector(s) of \(A^{-1}, A^n, A^T\)
Working¶
| Scenario | Method |
|---|---|
| Repeating eigen values | back substitution |
| else | Cramer’s rule for 2 independent rows |