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

Question

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.

Answer 1

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 $(\lambda_1-x)(\lambda_2-x)\ldots(\lambda_n-x)=(\mu_1-x)(\mu_2-x)\ldots(\mu_n-x)$, …