This is the way I use to remember which form to use. Note the T {\displaystyle T\,} matrix consists of all the eigenvectors, v i {\displaystyle v_{i}\,} , of A {\displaystyle A\,} : T = ( v 1 | v 2 | ⋯ | v n ) {\displaystyle T=(v_{1}|v_{2}|\cdots |v_{n})\,} . Using the definition of (right) eigenvectors: A v i = λ i v i {\displaystyle Av_{i}=\lambda _{i}v_{i}\,} , we have:
A T = A ( v 1 | v 2 | ⋯ | v n ) = ( A v 1 | A v 2 | ⋯ | A v n ) = ( λ 1 v 1 | λ 2 v 2 | ⋯ | λ n v n ) {\displaystyle AT=A(v_{1}|v_{2}|\cdots |v_{n})=(Av_{1}|Av_{2}|\cdots |Av_{n})=(\lambda _{1}v_{1}|\lambda _{2}v_{2}|\cdots |\lambda _{n}v_{n})\,} = ( v 1 | v 2 | ⋯ | v n ) ( λ 1 λ 2 ⋱ λ n ) = T Λ {\displaystyle =(v_{1}|v_{2}|\cdots |v_{n}){\begin{pmatrix}\lambda _{1}\\&\lambda _{2}\\&&\ddots \\&&&\lambda _{n}\end{pmatrix}}=T\Lambda \,}
Right multiply each side by T − 1 {\displaystyle T^{-1}\,} : A = T Λ T − 1 {\displaystyle A=T\Lambda T^{-1}\,} , which is the correct form.
--Shuo
matrix consists of all the eigenvectors, 
, of 
: 
. Using the definition of (right) eigenvectors: 
, we have:
: 
, which is the correct form.
