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In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose —that is, the element in the i -th row and j -th column is equal to the complex conjugate of the element in the j -th row and i -th column, for all indices i and j: is Hermitian {\displaystyle A {\text { is ...
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix is an matrix obtained by transposing and applying complex conjugation to each entry (the complex conjugate of being , for real numbers and ). There are several notations, such as or , [1] , [2] or (often in physics) .
In mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite. Sylvester's criterion states that a n × n Hermitian matrix M is positive-definite if and only if all the following matrices have a positive determinant: the upper left 1-by-1 corner of M, the upper left ...
Examples include (but not limited to) symmetric, skew-symmetric, and normal matrices. Null-Hermitian matrix A square matrix whose null space (or kernel) is equal to its conjugate transpose, N(A)=N(A *) or ker(A)=ker(A *). Synonym for kernel-Hermitian matrices. Examples include (but not limited) to Hermitian, skew-Hermitian matrices, and normal ...
Let A be an m × n matrix and k an integer with 0 < k ≤ m, and k ≤ n.A k × k minor of A, also called minor determinant of order k of A or, if m = n, the (n − k) th minor determinant of A (the word "determinant" is often omitted, and the word "degree" is sometimes used instead of "order") is the determinant of a k × k matrix obtained from A by deleting m − k rows and n − k columns.
The general form of an inner product on is known as the Hermitian form and is given by , = † = † ¯, where is any Hermitian positive-definite matrix and † is the conjugate transpose of . For the real case, this corresponds to the dot product of the results of directionally-different scaling of the two vectors, with positive scale factors ...
The Householder matrix has the following properties: it is Hermitian: =,; it is unitary: =,; hence it is involutory: =.; A Householder matrix has eigenvalues .To see this, notice that if is orthogonal to the vector which was used to create the reflector, then =, i.e., is an eigenvalue of multiplicity , since there are independent vectors orthogonal to .
The Pauli matrices are a vector of three 2×2 matrices that are used as spin operators. Given a unit vector in 3 dimensions, for example (a, b, c), one takes a dot product with the Pauli spin matrices to obtain a spin matrix for spin in the direction of the unit vector. The eigenvectors of that spin matrix are the spinors for spin-1/2 oriented ...