How can I prove or disprove this claim?
Since $X$ is invertible, $X$'s eigenvalue $\lambda_i \neq 0$ for all $i = 1, \cdots ,n$. However, according to Wikipedia https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix, we need $n$ linearly independent eigenvectors in addition to $X$ being square in order to say whether or not $X$ has spectral decomposition. But we know $X$ is invertible, therefore $X$ has full rank meaning that all columns of $X$ are linearly independent, but doesn't mean we have $n$ linearly independent eigenvectors? So to me, I don't know invertible and square matrix are enough to say that $X$ has a spectral decomposition.