Exercise 1.6.17

If the columns of $X$, eigenvectors of $A$, are linearly independent, then

• 1.

  $A$ is invertible: False, we need the columns of $A$ are independent

• 1.

  $A$ is diagonalizable: True (wrong answer: False, we may have Geometric multiplicity < Algebraic multiplicity, so $A$ is not diagonalizable., well we already have $X$)

• 1.

  $X$ is invertible: True

• 1.

  $X$ is diagonalizable: False, we need the eigenvectors of $X$ now (wrong answer: True. for any $x$, we have $\mathit{Xx}=x$, the eigenvalues of $X$ are 1, and the diagonal matrix is $I$ for $X$).