I keep getting mixed up on whether the diagonalized form of A is T^(-1)AT or TAT^(-1). Is there an easy way to remember the correct form?

From MurrayWiki
Jump to: navigation, search

This is the way I use to remember which form to use. Note the T\, matrix consists of all the eigenvectors, v_{i}\,, of A\,: T=(v_{1}|v_{2}|\cdots |v_{n})\,. Using the definition of (right) eigenvectors: Av_{i}=\lambda _{i}v_{i}\,, we have:

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}|\cdots |v_{n}){\begin{pmatrix}\lambda _{1}\\&\lambda _{2}\\&&\ddots \\&&&\lambda _{n}\end{pmatrix}}=T\Lambda \,

Right multiply each side by T^{{-1}}\,: A=T\Lambda T^{{-1}}\,, which is the correct form.

--Shuo