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Hermitian matrices also appear in techniques like singular value decomposition (SVD) and eigenvalue decomposition. In statistics and machine learning, Hermitian matrices are used in covariance matrices, where they represent the relationships between different variables. The positive definiteness of a Hermitian covariance matrix ensures the well ...
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 ).
In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector, where is the row vector transpose of . [1] More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector , where denotes the ...
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 ...
A square complex matrix whose transpose is equal to the matrix with every entry replaced by its complex conjugate (denoted here with an overline) is called a Hermitian matrix (equivalent to the matrix being equal to its conjugate transpose); that is, A is Hermitian if = ¯.
When is a positive-semidefinite Hermitian matrix, and are both equal to the unitary matrix used to diagonalize . However, when M {\displaystyle \mathbf {M} } is not positive-semidefinite and Hermitian but still diagonalizable , its eigendecomposition and singular value decomposition are distinct.
If V is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of A is a Hermitian matrix, i.e., equal to its conjugate transpose A ∗. By the finite-dimensional spectral theorem, V has an orthonormal basis such that the matrix of A relative to this basis is a diagonal matrix with entries in the ...
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma ( σ ), they are occasionally denoted by tau ( τ ) when used in connection with isospin symmetries.