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Subject Item
dbr:Normal_matrix
rdfs:label
Normal matrix
rdfs:comment
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.
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dbr:Positive_real_numbers dbr:Orthogonal_matrix dbr:Real_number dbr:Normal_operator dbr:Idempotent_matrix dbr:Skew-Hermitian_matrix dbr:Complex_number dbr:Hermitian_matrix dbr:Unit_circle dbr:Ordinary_least_squares dbr:Complex_conjugate dbr:Eigenvalues_and_eigenvectors dbr:Spectral_theorem dbr:C*-algebra dbr:Invertible_matrix dbr:Skew-symmetric_matrix dbr:Involutory_matrix dbr:Unitary_matrix dbr:Diagonal_matrix dbr:Normed_vector_space dbr:Singular_value dbr:Matrix_norm dbr:Singular_value_decomposition dbr:Square_matrix dbr:Schur_decomposition dbr:Zero_matrix dbr:Quasinormal_operator dbr:Definite_matrix dbr:Commutative_property dbr:Polar_decomposition dbr:Conjugate_transpose dbr:Scaling_(geometry) dbr:Identity_matrix dbr:Symmetric_matrix dbr:Commuting_matrices dbr:Matrix_similarity dbr:Orthogonality dbr:Orthonormal_basis dbr:Imaginary_number dbr:Self-adjoint dbr:Diagonalizable_matrix dbr:Triangular_matrix
dbo:abstract
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. 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.
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