Conjugated invertible matrices/Invariant properties/Exercise

Let ${}\operatorname {GL} _{n}\!{\left(K\right)}$ be the set of all invertible ${}n\times n$-matrices over a field ${}K$. Show that for matrices ${}M$ and ${}N$ from ${}\operatorname {GL} _{n}\!{\left(K\right)}$ that are conjugated to each other, the following properties and invariants coincide: The determinant, the eigenvalues, the dimension of the eigenspace of an eigenvalue, the diagonalizability, the trigonalizability.

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