Analyzing the Given Matrix

The question asks us to classify the given matrix:

\( A = \left[ {\begin{array}{c} 0&{ - 4 + i}\\ {4 + i}&0 \end{array}} \right] \)

We need to determine if this matrix is symmetric, skew-symmetric, Hermitian, or Skew-Hermitian. Since the matrix contains complex numbers (the imaginary unit \(i\)), we should pay close attention to the definitions involving complex conjugates.

Understanding Matrix Properties

Let's define the relevant properties for complex matrices:

• Symmetric Matrix: A square matrix \(A\) is symmetric if \(A^T = A\), where \(A^T\) is the transpose of \(A\).
• Skew-Symmetric Matrix: A square matrix \(A\) is skew-symmetric if \(A^T = -A\).
• Hermitian Matrix: A square matrix \(A\) is Hermitian if \(A^\dagger = A\), where \(A^\dagger\) is the conjugate transpose (or adjoint) of \(A\). The conjugate transpose \(A^\dagger\) is obtained by taking the transpose of the complex conjugate of \(A\), i.e., \(A^\dagger = (A^*)^T\) or \(A^\dagger = (A^T)^*\). The complex conjugate \(A^*\) is obtained by replacing each element \(a_{jk}\) with its complex conjugate \(\overline{a_{jk}}\).
• Skew-Hermitian Matrix: A square matrix \(A\) is Skew-Hermitian if \(A^\dagger = -A\).

For matrices with complex entries, the concepts of Hermitian and Skew-Hermitian are more fundamental than symmetric and skew-symmetric, as they involve conjugation which is crucial for complex numbers.

Calculating Matrix Transpose and Conjugate Transpose

Let's calculate the transpose and conjugate transpose of the given matrix \(A\).

The given matrix is:

\( A = \left[ {\begin{array}{c} 0&{ - 4 + i}\\ {4 + i}&0 \end{array}} \right] \)

Transpose (\(A^T\))

To find the transpose, we swap rows and columns:

\( A^T = \left[ {\begin{array}{c} 0&{4 + i}\\ { - 4 + i}&0 \end{array}} \right] \)

Comparing \(A^T\) with \(A\), we see they are not equal. \(A^T \ne A\). Also, \(A^T \ne -A\). So, the matrix is neither symmetric nor skew-symmetric in the standard sense applied to complex matrices.

Complex Conjugate (\(A^*\))

To find the complex conjugate, we replace each element \(a_{jk}\) with its complex conjugate \(\overline{a_{jk}}\). Recall that for a complex number \(z = x + iy\), its conjugate is \(\overline{z} = x - iy\).

• \(\overline{0} = 0\)
• \(\overline{-4 + i} = -4 - i\)
• \(\overline{4 + i} = 4 - i\)

So, the complex conjugate matrix \(A^*\) is:

\( A^* = \left[ {\begin{array}{c} \overline{0}&{\overline{- 4 + i}}\\ {\overline{4 + i}}&{\overline{0}} \end{array}} \right] = \left[ {\begin{array}{c} 0&{- 4 - i}\\ {4 - i}&0 \end{array}} \right] \)

Conjugate Transpose (\(A^\dagger\))

To find the conjugate transpose \(A^\dagger\), we take the transpose of \(A^*\):

\( A^\dagger = (A^*)^T = \left[ {\begin{array}{c} 0&{4 - i}\\ {- 4 - i}&0 \end{array}} \right] \)

Checking for Hermitian or Skew-Hermitian Properties

Is \(A\) Hermitian?

A matrix \(A\) is Hermitian if \(A^\dagger = A\).

We have \(A^\dagger = \left[ {\begin{array}{c} 0&{4 - i}\\ {- 4 - i}&0 \end{array}} \right]\) and \(A = \left[ {\begin{array}{c} 0&{ - 4 + i}\\ {4 + i}&0 \end{array}} \right]\).

Clearly, \(A^\dagger \ne A\). So, the matrix is not Hermitian.

Is \(A\) Skew-Hermitian?

A matrix \(A\) is Skew-Hermitian if \(A^\dagger = -A\).

Let's calculate \(-A\):

\( -A = -1 \times \left[ {\begin{array}{c} 0&{ - 4 + i}\\ {4 + i}&0 \end{array}} \right] = \left[ {\begin{array}{c} -1 \times 0&{-1 \times (- 4 + i)}\\ {-1 \times (4 + i)}&{-1 \times 0} \end{array}} \right] = \left[ {\begin{array}{c} 0&{4 - i}\\ {- 4 - i}&0 \end{array}} \right] \)

Now, let's compare \(A^\dagger\) and \(-A\):

\( A^\dagger = \left[ {\begin{array}{c} 0&{4 - i}\\ {- 4 - i}&0 \end{array}} \right] \)

\( -A = \left[ {\begin{array}{c} 0&{4 - i}\\ {- 4 - i}&0 \end{array}} \right] \)

We see that \(A^\dagger = -A\).

Therefore, the given matrix is a Skew-Hermitian matrix.

Conclusion

Based on our calculations, the given matrix satisfies the condition \(A^\dagger = -A\). This is the definition of a Skew-Hermitian matrix.

Revision Table: Matrix Properties

Additional Information on Matrix Classification

When dealing with matrices containing complex numbers, the concepts of Hermitian and Skew-Hermitian are generalizations of symmetric and skew-symmetric matrices from the real case. For a real matrix (where all entries are real numbers), the complex conjugate of any element is just the element itself (\(\overline{a_{jk}} = a_{jk}\)). In this case, \(A^* = A\), and thus \(A^\dagger = (A^*)^T = A^T\).

• If \(A\) is a real matrix, \(A^\dagger = A^T\).
  • The condition for Hermitian (\(A^\dagger = A\)) becomes \(A^T = A\), which is the definition of a real symmetric matrix.
  • The condition for Skew-Hermitian (\(A^\dagger = -A\)) becomes \(A^T = -A\), which is the definition of a real skew-symmetric matrix.

This shows that Hermitian and Skew-Hermitian matrices are the appropriate classifications when complex entries are involved, encompassing the real symmetric and skew-symmetric cases as special instances.