Are all Hermitian matrices normal?
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.
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.
- A positive definite matrix is a symmetric matrix with all positive eigenvalues. Note that as it's a symmetric matrix all the eigenvalues are real, so it makes sense to talk about them being positive or negative. Now, it's not always easy to tell if a matrix is positive definite.
- In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative; that is, it satisfies the condition AT = −A.
- 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.
- Invertible matrix. Non-square matrices (m-by-n matrices for which m ≠ n) do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. If A is m-by-n and the rank of A is equal to n, then A has a left inverse: an n-by-m matrix B such that BA = In.
- 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
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., .
- In mathematics, the conjugate transpose or Hermitian transpose of an m-by-n matrix A with complex entries is the n-by-m matrix A∗ obtained from A by taking the transpose and then taking the complex conjugate of each entry. (
- A unitary operator is a bounded linear operator U : H → H on a Hilbert space H that satisfies U*U = UU* = I, where U* is the adjoint of U, and I : H → H is the identity operator. Thus a unitary operator is a bounded linear operator which is both an isometry and a coisometry, or, equivalently, a surjective isometry.
- In mathematics, a self-adjoint operator on a finite-dimensional complex vector space V with inner product is a linear map A (from V to itself) that is its own adjoint: . Since an everywhere-defined self-adjoint operator is necessarily bounded, one needs be more attentive to the domain issue in the unbounded case.
Updated: 7th December 2019