Differences

This shows you the differences between two versions of the page.

--- mvg:lecture_02:start [2018/11/19 12:09] – [Eigenvalue Problem] admin
+++ mvg:lecture_02:start [2020/02/04 15:12] (current) – [Eigenvalue Problem] admin
@@ Line 25: / Line 25: @@
 Linear transformation scales vector $\mathbf{v}$, scaling factor is Eigen value.
-Set of Eigen values is spectrum: $\sigma(A) = \left{\lambda_1 \ldots \lambda_n\right}$
+Set of Eigen values is spectrum:
+$\sigma(A) = \{ \lambda_1 \ldots \lambda_n \} $
 $A\mathbf{v} = \lambda \mathbf{v}$
+$(A-\lambda \one) A = 0$
+$\left(A-\lambda \unity \right) A = 0$
+**Question:**
+If $P$ is invertible and $B = P^{-1}AP \Rightarrow \sigma(A) = \sigma(B)$. Why?
+==== Symmetric Matrices ====
+Real matrix with real Eigenvalues is related to symmetric matrix.
+$S^T = S$
+Positive semidefinite: $x^TSx \ge 0$
+Positive definite: $x^TSx > 0$
 ==== Vector Spaces, Keywords ====