Jordan normal form/Diagonalizable part/Same eigenvalues/Exercise

Let ${}\varphi \colon V\rightarrow V$ be a trigonalizable linear mapping, together with its canonical additive decomposition

${}\varphi =\varphi _{\rm {diag}}+\varphi _{\rm {nil}}\,$

in the sense of fact. Show that the eigenvalues of ${}\varphi$ and of ${}\varphi _{\rm {diag}}$ coincide. Does this also hold for their algebraic multiplicity? Does this also hold for their geometric multiplicity?

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