mB = Transpose[{{1, 1, 1}, {1, 0, -1}, {1, -2, 1}}]

mD = {{-1, 0, 0}, {0, 1, 0}, {0, 0, 1}}

mM = mB . mD . Inverse[mB]

mM . {1, 0, -1}

mM . {1, 2, 3}