a symmetric matrix is a square matrix that is equal to its transpose. Formally, A  is symmetric ⟺ A = A T . {\displaystyle A{\text{ is symmetric}}\iff...

 skew-symmetric ⟺ A T = − A . {\displaystyle A{\text{ skew-symmetric}}\quad \iff \quad A^{\textsf {T}}=-A.} In terms of the entries of the matrix, if a...

In mathematics, a symmetric matrix   M   {\displaystyle \ M\ } with real entries is positive-definite if the real number   x ⊤ M x   {\displaystyle \ \mathbf...

= AT, is a symmetric matrix. If instead, A is equal to the negative of its transpose, that is, A = −AT, then A is a skew-symmetric matrix. In complex...

thus pi j = pj i, and the product matrix (pi j) is symmetric. Similarly, the product AT A is a symmetric matrix. A quick proof of the symmetry of A...

be understood as the complex extension of real symmetric matrices. If the conjugate transpose of a matrix A {\displaystyle A} is denoted by A H , {\displaystyle...

which maps any skew-symmetric matrix A to a rotation matrix. In fact, aside from the noted exceptions, we can produce any rotation matrix in this way. Although...

identity is also a permutation matrix. A reflection is its own inverse, which implies that a reflection matrix is symmetric (equal to its transpose) as well...

A^{*}A=AA^{*}} . If a real square matrix is symmetric, skew-symmetric, or orthogonal, then it is normal. If a complex square matrix is Hermitian, skew-Hermitian...

1s on the diagonal. Real symmetric matrices are diagonalizable by orthogonal matrices; i.e., given a real symmetric matrix A {\displaystyle A} , Q T...