load("fourie")$

conj_t(mat) := transpose(conjugate(mat))$

is_hermitian(mat) := equalp(mat, conj_t(mat))$
is_normal(mat) := equalp(mat.conj_t(mat), conj_t(mat).mat)$
is_unitary(mat) := equalp(ident(length(mat)), mat.conj_t(mat))$

mat1: matrix([3, 2+%i], [2-%i, 1])$

mat2: matrix([1, 1, 0], [0, 1, 1], [1, 0, 1])$

s: sqrt(2)/2$
mat3: matrix([s, s, 0], [-s*%i, s*%i, 0], [0, 0, %i])$

conj_tests(mat) := block(
    disp("Chosen matrix:", mat, "Conjugate transpose:", conj_t(mat)),
    test_funs: [is_hermitian, is_normal, is_unitary],
    queries: ["Hermitian? ", "Normal? ", "Unitary? "],
    for i from 1 thru 3 do(print(sconcat(queries[i], test_funs[i](mat))))
)$

map(conj_tests, [mat1, mat2, mat3]);