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If I have a matrix with distinct eigenvalues, why can't I just write a matrix with those as diagonal elements, but I have to do the procedure $P^{-1}AP$?

Also I have found this question too that seems to support what I say, but then why in books there is this whole procedure?

J. W. Tanner
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Adam
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    It depends what the question is. If you are just asked for the diagonal matrix then, yes, simply write it down. If you are asked to "diagonalise" the matrix, that usually means find $P$ as well. – David Feb 08 '19 at 00:56
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    A typical use case is to calculate $A^n$ for some big $n$. To answer this, diagonalize: $A=PDP^{-1}$ and then $A^n=PD^nP^{-1}$, where knowing $P$ is required. Note also that the columns of $P$ are just the eigenvectors. – Berci Feb 08 '19 at 01:02

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A diagonalization of a matrix $A$ is a decomposition as $A=PDP^{-1}$ for $P$ a change of basis matrix, $D$ a diagonal matrix, and $A$ the original matrix. Suppose we know that $$ \mathbb{R}^n=\bigoplus_{i=1}^kE(\lambda_i,A)$$ where $E(\lambda_i,A)$ is the eigenspace of $A$ corresponding to $\lambda_i$, and $\lambda_1,\ldots,\lambda_k$ are the distinct eigenvalues. Then we can immediately recover the diagonal matrix $D$ as having a diagonal consisting of $\lambda_1,\ldots \lambda_k$ with $\lambda_i$ appearing $\dim E(\lambda_i,A)$ times. However, in practice you won't have this much information.

Long story short its mostly a matter of what information you need to gather and what information you have. Sometimes you'll be more interested in the matrix $P$.

Alekos Robotis
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