Can a matrix be Hermitian and unitary?
Normal, Hermitian, and unitary matrices. A square matrix is a Hermitian matrix if it is equal to its complex conjugate transpose . If a Hermitian matrix is real, it is a symmetric matrix, . is a unitary matrix if its conjugate transpose is equal to its inverse , i.e., .
Skew-Hermitian Matrix. A square matrix, A , is skew-Hermitian if it is equal to the negation of its complex conjugate transpose, A = -A' . In terms of the matrix elements, this means that. a i , j = − a ¯ j , i . The entries on the diagonal of a skew-Hermitian matrix are always pure imaginary or zero.
- In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix A is symmetric if. Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal.
- In mathematics, a complex square matrix U is unitary if its conjugate transpose U∗ is also its inverse—that is, if. where I is the identity matrix. In physics, especially in quantum mechanics, the Hermitian conjugate of a matrix is denoted by a dagger (†) and the equation above becomes.
- Nilpotent Matrix. There are two equivalent definitions for a nilpotent matrix. A square matrix whose eigenvalues are all 0. 2. A square matrix such that is the zero matrix for some positive integer matrix power , known as the index (Ayres 1962, p. 11).
As a result of this definition, the diagonal elements of a Hermitian matrix are real numbers (since ), while other elements may be complex. Hermitian matrices have real eigenvalues whose eigenvectors form a unitary basis. For real matrices, Hermitian is the same as symmetric.
- The Multiplicative Identity. The identity property of multiplication states that when 1 is multiplied by any real number, the number does not change; that is, any number times 1 is equal to itself. The number "1" is called the multiplicative identity for real numbers.
- If a matrix has m rows and n columns we say it is an mxn matrix or that the SIZE of the matrix is mxn. In the example above, A is a 3x2 matrix. When m = n we say the matrix is SQUARE. In the example above, B is a square matrix and has size 2x2. The numbers in the array are called ELEMENTS or ENTRIES of the matrix.
- A geometric interpretation can be given to the value of the determinant of a square matrix with real entries: the absolute value of the determinant gives the scale factor by which area or volume (or a higher-dimensional analogue) is multiplied under the associated linear transformation, while its sign indicates whether
The normal matrices are the matrices which are unitarily diagonalizable, i.e., is a normal matrix iff there exists a unitary matrix such that is a diagonal matrix. All Hermitian matrices are normal but have real eigenvalues, whereas a general normal matrix has no such restriction on its eigenvalues.
- Among complex matrices, all unitary, Hermitian, and skew-Hermitian matrices are normal. Likewise, among real matrices, all orthogonal, symmetric, and skew-symmetric matrices are normal. However, it is not the case that all normal matrices are either unitary or (skew-)Hermitian.
- In quantum physics, you'll often work with Hermitian adjoints. The Hermitian adjoint — also called the adjoint or Hermitian conjugate — of an operator A is denoted. To find the Hermitian adjoint, you follow these steps: Replace complex constants with their complex conjugates.
- The normal matrices are the matrices which are unitarily diagonalizable, i.e., is a normal matrix iff there exists a unitary matrix such that is a diagonal matrix. All Hermitian matrices are normal but have real eigenvalues, whereas a general normal matrix has no such restriction on its eigenvalues.
Updated: 20th August 2018