Let V be the real vector space of all 2 times 2 complex Herm

Let V be the real vector space of all 2 times 2 (complex) Hermitian matrices, that is, 2 times 2 complex matrices A which satisfy A_ij = A_ji^bar. Show that the equation q (A) = det A defines a quadratic form g on V. Let W be the subspace of V of matrices of trace 0. Show that the bilinear form determined by q is negative definite on the subspace W.

Solution

Let the entries of the matrix A are : R₁    :   a b+ ic A is a Hermitian matrix

R2   : b-ic d

q(A) = det A = ad - (b+ic) (b-ic) = ad - (b² +c²) which a quadratic form on V

b . W = { B / trace of B =0}

Ist row R₁ : a b+ic

2nd row : b-ic -a as the trace = sum of the terms in the main diagonal

det B = - a² - (b+ic) (b-ic)

= - a² - b² -c² = - (a²+b²+c²) which is negative for all entries of the matrix \'B\"