  We construct $\ell $-spin-flipping matrices from the coefficient matrices of
pure states of $n$ qubits and show that the $\ell $-spin-flipping matrices are
congruent and unitary congruent whenever two pure states of $n$ qubits are
SLOCC and LU equivalent, respectively. The congruence implies the invariance of
ranks of the $\ell $-spin-flipping matrices under SLOCC and then permits a
reduction of SLOCC classification of n qubits to calculation of ranks of the
$\ell $-spin-flipping matrices. The unitary congruence implies the invariance
of singular values of the $\ell $-spin-flipping matrices under LU and then
permits a reduction of LU classification of n qubits to calculation of singular
values of the $\ell $-spin-flipping matrices. Furthermore, we show that the
invariance of singular values of the $\ell $-spin-flipping matrices $\Omega
_{1}^{(n)}$ implies the invariance of the concurrence for even $n$ qubits and
the invariance of the n-tangle for odd $n$ qubits. Thus, the concurrence and
the n-tangle can be used for LU classification and computing the concurrence
and the n-tangle only performs additions and multiplications of coefficients of
states.
