A hermitian matrix is a square matrix that is equal to its own conjugate transpose. This means that the elements along the main diagonal are real numbers, and the elements above and below the diagonal are complex conjugates of each other.
Hermitian matrices have many important properties and applications in mathematics and physics. They are used in quantum mechanics, signal processing, and optimization problems. They also have many nice properties that make them easier to work with in calculations.
A matrix can be tested for hermitian properties by checking if it is equal to its own conjugate transpose. This means that the matrix A must satisfy the equation A = A*, where A* is the conjugate transpose of A. If this equation is satisfied, then the matrix is hermitian.
The eigenvalues of a hermitian matrix are always real numbers. This is because the eigenvalues are the roots of the characteristic polynomial of the matrix, and for hermitian matrices, this polynomial has only real coefficients.
No, a non-square matrix cannot be hermitian. Hermitian matrices must be square, meaning they have the same number of rows and columns. This is because the conjugate transpose operation only applies to square matrices.