: Mathématiques: symmetric matrix

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Terme préférentiel

symmetric matrix

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
${\\displaystyle A{\ ext{ is symmetric}}\\iff A=A^{\ extsf {T}}.}$
Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if ${\\displaystyle a_{ij}}$ denotes the entry in the ${\\displaystyle i}$ th row and ${\\displaystyle j}$ th column then
${\\displaystyle A{\ ext{ is symmetric}}\\iff {\ ext{ for every }}i,j,\\quad a_{ji}=a_{ij}}$
for all indices ${\\displaystyle i}$ and ${\\displaystyle j.}$ Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. In linear algebra, a real symmetric matrix represents a self-adjoint operator represented in an orthonormal basis over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries. Symmetric matrices appear naturally in a variety of applications, and typical numerical linear algebra software makes special accommodations for them.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Symmetric_matrix)