# 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?

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Jump to navigationJump to searchThis 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\[ AT = A(v_1|v_2|\cdots|v_n) = (Av_1|Av_2|\cdots|Av_n) = (\lambda_1v_1|\lambda_2v_2|\cdots|\lambda_nv_n) \,\] \(= (v_1|v_2|\cdots|v_n)\begin{pmatrix} \lambda_1 \\ & \lambda_2 \\ & & \ddots \\ & & & \lambda_n \end{pmatrix}=T\Lambda \,\)

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

--Shuo