The eigenvectors of $A$ or column vectors of $X$ are linearly independent

Question

Let $X$ be the matrix whose columns are eigenvectors of the matrix $A$. If the eigenvectors of $A$ or column vectors of $X$ are linearly independent, then

$(i)$ $A$ is invertible

$(ii)$ $A$ is diagonalisable

$(iii)$ $X$ is invertible

$(iv)$ $X$ is diagonalisable.

Answer:

Since the eigenvectors of $A$ are linearly independent, $A$ is diagonalisable.

i.e., $A=XDX^{-1}$, where $D$ is the diagonal matrix of eigen values of $A$.

So $(ii)$ is true.

Also since the eigenvectors of $A$ are linearly independent , the matrix $A$ is invertible.

Hence $(i)$ is also true.

But how to decide about the option $(iii)$ and $(iv)$?

Help me

Answer 1

Guide:

• Let $A$ be the zero matrix, then $A$ is not invertible but it is diagonalizable.
• If $X$ is a square matrix and each column is linearly independent, then it is of full rank and hence it is invertible.
• Let $X= \begin{bmatrix} 1 & 1 \\ 0 & 1\end{bmatrix}$, then it is invertible but not diagonalizable.

Answer 2

HINT

Note that for $(iii)$ $X$ is also full rank.

For $(iv)$ it seems we do not have sufficient information.