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mvg:lecture_02:start [2018/11/19 12:09] – [Eigenvalue Problem] adminmvg:lecture_02:start [2020/02/04 15:12] (current) – [Eigenvalue Problem] admin
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 Linear transformation scales vector $\mathbf{v}$, scaling factor is Eigen value. 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 \} $
  
 $A\mathbf{v} = \lambda \mathbf{v}$ $A\mathbf{v} = \lambda \mathbf{v}$
 +
 +$(A-\lambda \one) A = 0$
 +
 +$\left(A-\lambda \unity \right) A = 0$
 +
 +**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 ==== ==== Vector Spaces, Keywords ====
  
mvg/lecture_02/start.1542629348.txt.gz · Last modified: 2018/11/19 12:09 by admin