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A complex symmetric matrix can be 'diagonalized' using a unitary matrix: thus if is a complex symmetric matrix, there is a unitary matrix such that is a real diagonal matrix with non-negative entries.
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 ...
By the definition of matrix equality, which requires that the entries in all corresponding positions be equal, equal matrices must have the same dimensions (as matrices of different sizes or shapes cannot be equal). Consequently, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main ...
Correlation matrix — a symmetric n×n matrix, formed by the pairwise correlation coefficients of several random variables. Covariance matrix — a symmetric n×n matrix, formed by the pairwise covariances of several random variables. Sometimes called a dispersion matrix. Dispersion matrix — another name for a covariance matrix.
A square matrix A that is equal to its transpose, that is, A = A T, is a symmetric matrix. If instead, A is equal to the negative of its transpose, that is, A = −A T, then A is a skew-symmetric matrix.
The identity matrix commutes with all matrices. Jordan blocks commute with upper triangular matrices that have the same value along bands. If the product of two symmetric matrices is symmetric, then they must commute. That also means that every diagonal matrix commutes with all other diagonal matrices. [9] [10] Circulant matrices commute.
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
A square diagonal matrix is a symmetric matrix, so this can also be called a symmetric diagonal matrix. The following matrix is square diagonal matrix: [] If the entries are real numbers or complex numbers, then it is a normal matrix as well.