# Eigendecomposition steps ¶

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The steps for the eigendecomposition are

• calculate $\lambda{I_n}-A_n$
• solve the lambdas (eigenvalues) in $det(\lambda{I_n}-A_n) = 0$
• you should have $n$ eigenvalues $\lambda_n$
• the order does not matter
• Now you got $D$, the eigenvalues are on the diagonal
• for each eigenvalue
• you can solve $A - \lambda_n * Id_n = 0$
• you can write it as a system
$\begin{split}\left \{ \begin{array}{r c l} (a_1 - \lambda_n) x + b_1 y + ... = 0 \\ a_2 x + (b_2 - \lambda_n) y + ... = 0 \\ ... \end{array} \right .\end{split}$
• each result (the vector (x,y, ...)) is an eigenvector
• by concatenating all of your eigenvectors, you get $P$
• solve $P^{-1}$
• check $P*D^1*P^{-1}=A$