Linear algebra matrix computations functions of matrices

mat = {{1., 2.}, {4., 5.}};
mat^0

{{1., 1.}, {1., 1.}}

MatrixExp[mat, 0]

MatrixExp[{{1., 2.}, {4., 5.}}, 0]

mat = {{1., 2.}, {5., 1.}};

MatrixPower[mat, 4]

{{161., 88.}, {220., 161.}}

matSq = mat.mat;matSq.matSq

{{161., 88.}, {220., 161.}}

MatrixExp[mat]

{{32.16634241136035, 20.271009131525236}, {50.67752282881309, 32.16634241136035}}

{eigs, vecs} = Eigensystem[mat];
vecs = Transpose[vecs];
vecsInv = Inverse[vecs];
vecs.DiagonalMatrix[Exp[eigs]].vecsInv

{{32.16634241136035, 20.271009131525236}, {50.67752282881308, 32.16634241136034}}