# Diagonalizable matrix is similar to a diagonal matrix with its eigenvalues as the diagonal entries

My book defines a diagonalizable matrix as follows:

A matrix $$A$$ is diagonalizable if it is similar to a diagonal matrix say $$D$$. So there exists an invertible matrix $$P$$ such that $$A =PDP^{-1}$$.

Now let eigen values of a diagonalizable matrix $$A$$ are $$\lambda_1, \lambda_2,\dots,\lambda_n$$.

How do I show that $$A$$ is similar to a diagonal matrix with $$\lambda_1, \lambda_2,\dots,\lambda_n$$ as its diagonal entries.

If $$A$$ and $$D$$ are similar, then they have the same characteristic polynomials. But the characteristic polynomial of $$A$$ is $$(\lambda_1-x)(\lambda_2-x)\ldots(\lambda_n-x)$$ and, if$$D=\begin{bmatrix}\mu_1&0&0&\ldots&0\\0&\mu_2&0&\ldots&0\\0&0&\mu_3&\ldots&0\\\vdots&\vdots&\vdots&\ddots&\vdots\\0&0&0&\ldots&\mu_n\end{bmatrix},$$then the characteristic polynomial of $$D$$ is $$(\mu_1-x)(\mu_2-x)\ldots(\mu_n-x)$$. Since $$(\lambda_1-x)(\lambda_2-x)\ldots(\lambda_n-x)=(\mu_1-x)(\mu_2-x)\ldots(\mu_n-x)$$, …