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--- mvg:lecture_02:start [2018/11/19 12:02] – [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:  $\sigma(A) = \{\lambda_1 \ldots \lambda_n}$
+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 ====