Analyzing Matrix Properties: Hermitian and Skew-Hermitian

This problem asks us to determine the properties of a given complex matrix \({\rm{A}}\) and a related matrix expression \({\rm{(\bar A}}{{\rm{)}}}^{\rm{T}}} + {\rm{A}}\). We need to check if they are Hermitian or skew-Hermitian.

First, let's understand the definitions:

• A square matrix \({\rm{M}}\) is Hermitian if it is equal to its conjugate transpose. The conjugate transpose of a matrix \({\rm{M}}\), denoted by \({\rm{M}}^{\dagger}\) or \({\rm{(\bar M}}{{\rm{)}}}^{\rm{T}}}\), is obtained by taking the complex conjugate of each element and then transposing the resulting matrix. So, \({\rm{M}}\) is Hermitian if \({\rm{M}} = {\rm{M}}^{\dagger}\).
• A square matrix \({\rm{M}}\) is Skew-Hermitian if it is equal to the negative of its conjugate transpose. So, \({\rm{M}}\) is Skew-Hermitian if \({\rm{M}} = -{\rm{M}}^{\dagger}\).

The given matrix is:

\({\rm{A}} = \left[ {\begin{array}{*{20}{c}} {1 - {\rm{i}}}&{\rm{i}}\\ { - {\rm{i}}}&{1 - {\rm{i}}} \end{array}} \right]\)

Step 1: Calculate the Conjugate Transpose of A

First, find the complex conjugate of A, denoted by \({\rm{\bar A}}\). We replace every \({\rm{i}}\) with \(-{\rm{i}}\) in the matrix A.

\({\rm{\bar A}} = \left[ {\begin{array}{*{20}{c}} {1 - (-\rm i)}&{-\rm i}\\ { - (-\rm i)}&{1 - (-\rm i)} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {1 + {\rm{i}}}&{-\rm i}\\ {\rm i}&{1 + {\rm{i}}} \end{array}} \right]\)

Next, find the transpose of \({\rm{\bar A}}\), which is \({\rm{(\bar A}}{{\rm{)}}}^{\rm{T}}}\) or \({\rm{A}}^{\dagger}\). We swap the rows and columns of \({\rm{\bar A}}\).

\({\rm{A}}^{\dagger} = {\rm{(\bar A}}{{\rm{)}}}^{\rm{T}}} = \left[ {\begin{array}{*{20}{c}} {1 + {\rm{i}}}&{\rm i}\\ {-\rm i}&{1 + {\rm{i}}} \end{array}} \right]\)

Step 2: Check if A is Hermitian or Skew-Hermitian

Compare A with \({\rm{A}}^{\dagger}\) to check if A is Hermitian:

\({\rm{A}} = \left[ {\begin{array}{*{20}{c}} {1 - {\rm{i}}}&{\rm{i}}\\ { - {\rm{i}}}&{1 - {\rm{i}}} \end{array}} \right]\)

\({\rm{A}}^{\dagger} = \left[ {\begin{array}{*{20}{c}} {1 + {\rm{i}}}&{\rm i}\\ {-\rm i}&{1 + {\rm{i}}} \end{array}} \right]\)

Since \({\rm{A}} \neq {\rm{A}}^{\dagger}\), the matrix A is not Hermitian.

Now, compare A with \(-{\rm{A}}^{\dagger}\) to check if A is Skew-Hermitian:

\(-{\rm{A}}^{\dagger} = -\left[ {\begin{array}{*{20}{c}} {1 + {\rm{i}}}&{\rm i}\\ {-\rm i}&{1 + {\rm{i}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {-1 - {\rm{i}}}&{-\rm i}\\ {\rm i}&{-1 - {\rm{i}}} \end{array}} \right]\)

Since \({\rm{A}} \neq -{\rm{A}}^{\dagger}\), the matrix A is not Skew-Hermitian.

Step 3: Calculate the Matrix \({\rm{B}} = {\rm{(\bar A}}{{\rm{)}}}^{\rm{T}}} + {\rm{A}}\)

Let \({\rm{B}} = {\rm{(\bar A}}{{\rm{)}}}^{\rm{T}}} + {\rm{A}}\). We add the matrices \({\rm{A}}^{\dagger}\) and \({\rm{A}}\) element by element:

\({\rm{B}} = \left[ {\begin{array}{*{20}{c}} {1 + {\rm{i}}}&{\rm i}\\ {-\rm i}&{1 + {\rm{i}}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {1 - {\rm{i}}}&{\rm{i}}\\ { - {\rm{i}}}&{1 - {\rm{i}}} \end{array}} \right]\)

\({\rm{B}} = \left[ {\begin{array}{*{20}{c}} {(1 + {\rm{i}}) + (1 - {\rm{i}})}&{{\rm i} + {\rm i}}\\ {-\rm i} + ({- \rm i})&{(1 + {\rm{i}}) + (1 - {\rm{i}})} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {2}&{2{\rm i}}\\ {-2{\rm i}}&{2} \end{array}} \right]\)

Step 4: Check if B is Hermitian or Skew-Hermitian

To check the property of B, we need to calculate its conjugate transpose, \({\rm{(\bar B}}{{\rm{)}}}^{\rm{T}}}\).

First, find the complex conjugate of B, \({\rm{\bar B}}\):

\({\rm{\bar B}} = \left[ {\begin{array}{*{20}{c}} {2}&{2(-\rm i)}\\ {-2(-\rm i)}&{2} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {2}&{-2{\rm i}}\\ {2{\rm i}}&{2} \end{array}} \right]\)

Next, find the transpose of \({\rm{\bar B}}\), which is \({\rm{(\bar B}}{{\rm{)}}}^{\rm{T}}}\):

\({\rm{(\bar B}}{{\rm{)}}}^{\rm{T}}} = \left[ {\begin{array}{*{20}{c}} {2}&{2{\rm i}}\\ {-2{\rm i}}&{2} \end{array}} \right]\)

Now, compare B with \({\rm{(\bar B}}{{\rm{)}}}^{\rm{T}}}\) to check if B is Hermitian:

\({\rm{B}} = \left[ {\begin{array}{*{20}{c}} {2}&{2{\rm i}}\\ {-2{\rm i}}&{2} \end{array}} \right]\)

\({\rm{(\bar B}}{{\rm{)}}}^{\rm{T}}} = \left[ {\begin{array}{*{20}{c}} {2}&{2{\rm i}}\\ {-2{\rm i}}&{2} \end{array}} \right]\)

Since \({\rm{B}} = {\rm{(\bar B}}{{\rm{)}}}^{\rm{T}}}\), the matrix \({\rm{B}} = {\rm{(\bar A}}{{\rm{)}}}^{\rm{T}}} + {\rm{A}}\) is Hermitian.

For completeness, let's check if B is Skew-Hermitian:

\(-{\rm{(\bar B}}{{\rm{)}}}^{\rm{T}}} = -\left[ {\begin{array}{*{20}{c}} {2}&{2{\rm i}}\\ {-2{\rm i}}&{2} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {-2}&{-2{\rm i}}\\ {2{\rm i}}&{-2} \end{array}} \right]\)

Since \({\rm{B}} \neq -{\rm{(\bar B}}{{\rm{)}}}^{\rm{T}}}\), the matrix B is not Skew-Hermitian.

Conclusion on Matrix Properties

Based on our calculations:

• Matrix A is neither Hermitian nor Skew-Hermitian.
• The matrix \({\rm{(\bar A}}{{\rm{)}}}^{\rm{T}}} + {\rm{A}}\) is Hermitian.

Therefore, the correct statement is that \({\rm{(\bar A}}{{\rm{)}}}^{\rm{T}}} + {\rm{A}}\) is hermitian.

Revision Table: Key Matrix Definitions

Additional Information on Hermitian and Skew-Hermitian Matrices

Here are some useful properties related to Hermitian and Skew-Hermitian matrices:

• For any square matrix A with complex entries, the matrix \({\rm{A}} + {\rm{A}}^{\dagger}\) is always Hermitian.
• For any square matrix A with complex entries, the matrix \({\rm{A}} - {\rm{A}}^{\dagger}\) is always Skew-Hermitian.
• Any square matrix A with complex entries can be uniquely expressed as the sum of a Hermitian matrix and a Skew-Hermitian matrix: \({\rm{A}} = {\rm{H}} + {\rm{S}}\), where \({\rm{H}} = \frac{1}{2}({\rm{A}} + {\rm{A}}^{\dagger})\) (Hermitian part) and \({\rm{S}} = \frac{1}{2}({\rm{A}} - {\rm{A}}^{\dagger})\) (Skew-Hermitian part).
• The diagonal elements of a Hermitian matrix are always real numbers.
• The diagonal elements of a Skew-Hermitian matrix are always purely imaginary or zero.

In this problem, we verified the property that \({\rm{A}} + {\rm{A}}^{\dagger}\) (which is \({\rm{(\bar A}}{{\rm{)}}}^{\rm{T}}} + {\rm{A}}\)) is Hermitian, which is a general property for any square matrix A.