If \(A = \left[ {\begin{array}{*{20}{c}} 0&{ - \;2 + i}\\ {2 - i}&0 \end{array}} \right] = \frac{1}{2} \cdot \left( {P + Q} \right)\) where P is hermitian and Q is skew hermitian matrix P and Q are ?

Concept:

Hermitian Matrix:

Any square matrix say A is said to be a hermitian matrix if A = Aθ where Aθ is the transpose of the conjugate matrix of A.

Note: All the diagonal elements of a hermitian matrix is purely real.

Skew hermitian Matrix:

Any square matrix say A is said to be a skew hermitian matrix if A = - Aθ where Aθ is the transpose of the conjugate matrix of A.

Note: All the diagonal elements of a skew hermitian matrix is purely imaginary or zero.

Every square matrix say A can be uniquely expressed as sum of hermitian and skew hermitian matrix as shown below:

\(A = \frac{1}{2} \cdot \left[ {\left( {A + {A^θ }} \right) + \left( {A - {A^θ }} \right)} \right]\) where (A + Aθ) and (A - Aθ) are hermitian and skew hermitian matrices respectively. 

Calculation: 

Given: \(A = \left[ {\begin{array}{*{20}{c}} 0&{ - \;2 + i}\\ {2 - i}&0 \end{array}} \right] = \frac{1}{2} \cdot \left( {P + Q} \right)\) where P is hermitian and Q is skew hermitian matrix

Here we have to find the matrix P and Q

As we know that, any square matrix can be be expressed as sum of hermitian and skew hermitian matrix.

i.e If A is a square matrix then A can be expressed as: \(A = \frac{1}{2} \cdot \left[ {\left( {A + {A^θ }} \right) + \left( {A - {A^θ }} \right)} \right]\) where (A + Aθ) and (A - Aθ) are hermitian and skew hermitian matrices respectively.