Conversion of a Hermitian Matrix into a Skew Hermitian Matrix

A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. In other words, the matrix is equal to the complex conjugate of its own transpose. A Skew Hermitian matrix, on the other hand, is a square matrix whose conjugate transpose is equal to the negative of the matrix itself.

To convert a Hermitian matrix into a Skew Hermitian matrix, we can use the following steps:

Step 1: Take the conjugate transpose of the Hermitian matrix
- The conjugate transpose of a matrix is obtained by taking the complex conjugate of each element and then transposing the matrix.
- Let's assume our Hermitian matrix is A. The conjugate transpose of A is denoted as A*.
- Mathematically, A* = (A)^H, where (A)^H represents the complex conjugate of A.

Step 2: Multiply the conjugate transpose by 'i'
- In order to convert the Hermitian matrix into a Skew Hermitian matrix, we need to multiply the conjugate transpose by 'i', which is the imaginary unit.
- Mathematically, B = i * A*, where B is the resulting Skew Hermitian matrix.

Step 3: Verify if B is a Skew Hermitian matrix
- To verify if B is a Skew Hermitian matrix, we need to check if the conjugate transpose of B is equal to the negative of B.
- If (B)^H = -B, then B is a Skew Hermitian matrix.

Explanation of the Correct Answer
The correct answer to the given question is option 'B' (i) because when we multiply the conjugate transpose of a Hermitian matrix by 'i', we obtain a Skew Hermitian matrix.
- Option 'A' (-1) is incorrect because multiplying the Hermitian matrix by -1 would not result in a Skew Hermitian matrix.
- Option 'C' (i-d) is incorrect because subtracting a complex number from 'i' would not result in a Skew Hermitian matrix.
- Option 'D' (None of these) is incorrect because the correct answer is option 'B' (i), as explained above.

Therefore, to convert a Hermitian matrix into a Skew Hermitian matrix, the Hermitian matrix must be multiplied by 'i'.