In mathematics, a complex square matrix A is normal if it commutes with its conjugate transpose A*:
The concept of normal matrices can be extended to normal operators on infinite-dimensional normed spaces and to normal elements in C*-algebras. As in the matrix case, normality means commutativity is preserved, to the extent possible, in the noncommutative setting. This makes normal operators, and normal elements of C*-algebras, more amenable to analysis.
The spectral theorem states that a matrix is normal if and only if it is unitarily similar to a diagonal matrix, and therefore any matrix A satisfying the equation A*A = AA* is diagonalizable. (The converse does not hold because diagonalizable matrices may have non-orthogonal eigenspaces.) Thus and where is a diagonal matrix whose diagonal values are in general complex.
The left and right singular vectors in the singular value decomposition of a normal matrix differ only in complex phase from each other and from the corresponding eigenvectors, since the phase must be factored out of the eigenvalues to form singular values.
complex square matrix A is normal if it commutes with its conjugate transpose A*: A normal ⟺ A ∗ A = A A ∗ . {\displaystyle A{\text{ normal}}\iff A^{*}A=AA^{*}...
mathematics, a square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n {\displaystyle...
the matrixnormal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution...
.} The matrix X T X {\displaystyle \mathbf {X} ^{\operatorname {T} }\mathbf {X} } is known as the normalmatrix or Gram matrix and the matrix X T y {\displaystyle...
{T} }} is called a centered normal random vector if there exists a deterministic k × ℓ {\displaystyle k\times \ell } matrix A {\displaystyle {\boldsymbol...
matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition"...
algebra, a Jordan normal form, also known as a Jordan canonical form, is an upper triangular matrix of a particular form called a Jordan matrix representing...
entries of a matrix. Anti-diagonal matrix Banded matrix Bidiagonal matrix Diagonally dominant matrix Diagonalizable matrix Jordan normal form Multiplication...
matrix A, one is interested in finding a simple "normal form" B which is similar to A—the study of A then reduces to the study of the simpler matrix B...
algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions;...
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...
y⟩. U is normal ( U ∗ U = U U ∗ {\displaystyle U^{*}U=UU^{*}} ). U is diagonalizable; that is, U is unitarily similar to a diagonal matrix, as a consequence...
(conjugate transpose) of Q, and therefore normal (Q∗Q = QQ∗) over the real numbers. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation...
matrix A {\displaystyle A} . A (real-valued) symmetric matrix is necessarily a normalmatrix. Denote by ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot...
M (so N is self-adjoint). A normalmatrix is the matrix expression of a normal operator on the Hilbert space Cn. Normal operators are characterized by...
linear algebra, a square matrix A {\displaystyle A} is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there exists...
Any normalmatrix is similar to a diagonal matrix, since its Jordan normal form is diagonal. Eigenvectors of distinct eigenvalues of a normalmatrix are...
rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix R = [...
Gram matrix G {\displaystyle G} is also a normalmatrix. The Gram matrix of any orthonormal basis is the identity matrix. Equivalently, the Gram matrix of...
move through the scene at normal speed, allowing the sped-up movements of certain characters to be perceived normally. The Matrix opened in theaters in the...
functions Normal function, in set theory Normal invariants, in geometric topology Normalmatrix, a matrix that commutes with its conjugate transpose Normal measure...
In mathematics, the Smith normal form (sometimes abbreviated SNF) is a normal form that can be defined for any matrix (not necessarily square) with entries...
In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column...
normal. The complex normal family has three parameters: location parameter μ, covariance matrix Γ {\displaystyle \Gamma } , and the relation matrix C...
transposition. An m by n matrix A with integer entries has a (row) Hermite normal form H if there is a square unimodular matrix U where H=UA and H has the...
real symmetric matrix and more generally a Hermitian matrix, and a unitary matrix, is never defective; more generally, a normalmatrix (which includes...
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