Verify that the eigenvalues of the Hermitian matrix A are...

Verify that the eigenvalues of the Hermitian matrix $A$ are real and that eigenvectors from different eigenspaces are orthogonal (see Theorem 7.5 .2 ).
$$A=\left[\begin{array}{cc}
0 & 2 i \\
-2 i & 2
\end{array}\right]$$

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Key Concepts

Hermitian Matrix

A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. This property implies that all the entries on the main diagonal are real, and the off-diagonal entries are complex conjugates of each other. Hermitian matrices play a crucial role in many areas of mathematics and physics because they possess desirable spectral properties.

Real Eigenvalues

One of the fundamental properties of Hermitian matrices is that all their eigenvalues are real. This is due to the fact that for any eigenvalue and corresponding eigenvector, the expression involving the matrix and its conjugate transpose forces the eigenvalue to equal its own complex conjugate, which can only hold if the eigenvalue is real.

Orthogonal Eigenvectors

Another important feature of Hermitian matrices is that eigenvectors corresponding to distinct eigenvalues are orthogonal. This orthogonality arises from the symmetric structure of the matrix (when considering its conjugate transpose) and is a key property used in many applications such as quantum mechanics and numerical linear algebra.

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