14、Matrix Decomposition Theorems: Schur and Jordan

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于 2025-10-26 12:01:01 发布

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                     Matrix Decomposition Theorems: Schur and Jordan  
 1. Schur’s Decomposition Theorem and Related Results  
 1.1 Necessary and Sufficient Conditions for Diagonal Reduction  
  Non - singular Matrix Case  : A nonsingular matrix (T) exists such that (T^{-1}AT=\Lambda) (diagonal) if and only if there is a set of (n) linearly independent vectors, each of which is an eigenvector of (A). 
    Proof  : If (A) has (n) linearly independent eigenvectors (x_1,\cdots,x_n), let (T=(x_1,\cdots,x_n)). Then (AT = A(x_1,\cdots,x_n)=(Ax_1,\cdots,Ax_n)=(\lambda_1x_1,\cdots,\lambda_nx_n)=(x_1,\cdots,x_n)\Lambda = T\Lambda), so (T^{-1}AT=\Lambda). Conversely, if (AT = T\Lambda), then (A(T e_i)=(AT)e_i=(T\Lambda)e_i=T(\Lambda e_i)