Great. Thanks.Dick said:Yes, the answers to A and B are 'yes'. And yes, a hermitian nxn matrix has n (not necessarily distinct) eigenvalues. In the sense that it's characteristic equation has n real linear factors.
Ahh, I see. This I found from Wikipedia (http://en.wikipedia.org/wiki/Richard):Dick said:'Dick' is a nickname for 'Richard' (just like 'Rick'). I have no idea where the 'D' came from, it's just what people call me.
A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. In other words, if A is a Hermitian matrix, then A is equal to A*, where A* is the conjugate transpose of A.
A Hermitian matrix is a generalization of a symmetric matrix in complex numbers. While a symmetric matrix is equal to its own transpose, a Hermitian matrix is equal to its own conjugate transpose.
Some key properties of Hermitian matrices include:
- All eigenvalues are real
- Eigenvectors corresponding to distinct eigenvalues are orthogonal
- The trace of a Hermitian matrix is equal to the sum of its eigenvalues
- All submatrices formed by selecting rows and columns of a Hermitian matrix are also Hermitian matrices
Hermitian matrices are used in linear algebra for a variety of applications, including:
- Diagonalization of matrices
- Quadratic forms
- Inner product spaces
- Orthonormal bases
- Unitary matrices
- Spectral theorem
No, a non-square matrix cannot be a Hermitian matrix. Hermitian matrices are always square matrices with the same number of rows and columns.