Question

For a 4 x 4 matrix \(M\isin M_4(\mathbb{C})\), let M denote the matrix obtained from M by replacing each entry of M by its complex conjugate. Consider the real vector space
H = \(\{M\isin M_4(\mathbb{C}):M^T=\overline M\)
where M^T denotes the transpose of M. The dimension of H as a vector space over \(\R\) is equal to

• A. 6
• B. 16
• C. 15
• D. 12

Answer 1

The Correct Option is B

To find the dimension of the real vector space \(H\) given by:

\[ H = \{ M \in M_4(\mathbb{C}) : M^T = \overline{M} \} \]

where \(M^T\) denotes the transpose of the matrix \(M\) and \(\overline{M}\) denotes the matrix obtained by taking the complex conjugate of each entry, we need to find matrices that satisfy \(M^T = \overline{M}\).

1. First, consider that a \(4 \times 4\) complex matrix \(M\) has \(4 \times 4 = 16\) complex entries. Each complex entry can be written as \(a + bi\), where \(a\) and \(b\) are real numbers.
2. Hence, each complex entry can be described by two real numbers, so a \(4 \times 4\) matrix with complex entries is equivalent to a \(16 \times 2 = 32\)-dimensional real vector space.
3. The condition \(M^T = \overline{M}\) implies that for each pair \((i, j)\), the entry \((i, j)\) in \(M\) satisfies \((M)_{ij} = \overline{(M)_{ji}}\). Thus:
4. Diagonal entries \((i = j)\) of \(M\) must be real because they satisfy simultaneously \( (M)_{ii} = \overline{(M)_{ii}} \), meaning they equal their own complex conjugate.
5. For diagonal entries, there are \(4\) real numbers satisfying these conditions, one for each diagonal entry of a \(4 \times 4\) matrix.
6. For off-diagonal pairs \((i \neq j)\), each requires:
   (M)_{ij} = \overline{(M)_{ji}},\ (\forall i\neq j)
7. Thus, for each off-diagonal pair, you have one complex number that determines the other by complex conjugation. Since each pair thus adds \((1 + 1) = 2\) real dimensions, consider the combination \(ij\) and \(ji\) only once for each pair, meaning there are \(\frac{4 \times (4 - 1)}{2} = 6\) independent choices providing \(2 \times 6 = 12\) dimensions contributed from off-diagonal choices.
8. Adding these dimensions gives us \(4\) (from the diagonal) + \(12\) (from off-diagonal pairs) = \(16\).

Hence, the dimension of \(H\) as a vector space over \(\mathbb{R}\) is \( \boxed{16} \).