Eigen vectors of the matrix whose columns are eigen vectors of the original matrix

Question

Consider a matrix $A$ of dimension $n$X$n$ whose eigen vectors are $y_1,y_2,y_3,...,y_n$ and are linearly independent. What are the properties of the eigen vectors of the matrix $P$ whose columns are $y_1$,$y_2$,$y_3$,....,$y_n$ ? Of course, we know that $AP=PK$ where $K$ is the diagonal matrix containing the eigen values of $A$. Can we say anything about the linear independence of the eigen vectors of $P$ for its diagonalizability?

Thanks, Aravind.

Answer 1

$P$ can be any invertible matrix.

Let $P\in GL_n(\mathbb{F})$, and call the columns of $P$ $y_1,\ldots,y_n$. Then $\{y_1,\ldots,y_n\}$ is a basis of $\mathbb{F}^n$, and we can define a linear transformation $T:\mathbb{F}^n\to\mathbb{F}^n$ by $y_i\mapsto\lambda_iy_i$, where $\lambda_i$ are scalars. Represent $T$ by a matrix $A$, and by the above process one can obtain $P$ (among many other matrices).