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Every real symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix. If and are real symmetric matrices that commute, then they can be simultaneously diagonalized by an orthogonal matrix: [2] there exists a basis of such that every element of the basis is an eigenvector for both and . Every real symmetric matrix is ...
A symmetric matrix can always be transformed in this way into a diagonal matrix which has only entries , + , along the diagonal. Sylvester's law of inertia states that the number of diagonal entries of each kind is an invariant of A {\displaystyle A} , i.e. it does not depend on the matrix S {\displaystyle S} used.
Every square diagonal matrix is symmetric, since all off-diagonal entries are zero. Similarly, 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 over a real inner product space.
where denotes the transpose of and is a fixed nonsingular, skew-symmetric matrix.This definition can be extended to matrices with entries in other fields, such as the complex numbers, finite fields, p-adic numbers, and function fields.
A symmetric real matrix A is called positive-definite if the associated quadratic form = has a positive value for every nonzero vector x in . If f (x) ...
Bisymmetric matrices are both symmetric centrosymmetric and symmetric persymmetric.; The product of two bisymmetric matrices is a centrosymmetric matrix. Real-valued bisymmetric matrices are precisely those symmetric matrices whose eigenvalues remain the same aside from possible sign changes following pre- or post-multiplication by the exchange matrix.
The Courant minimax principle, as well as the maximum principle, can be visualized by imagining that if ||x|| = 1 is a hypersphere then the matrix A deforms that hypersphere into an ellipsoid. When the major axis on the intersecting hyperplane are maximized — i.e., the length of the quadratic form q ( x ) is maximized — this is the ...
The matrix A in the above decomposition is a symmetric matrix. In particular, by the spectral theorem, it has real eigenvalues and is diagonalizable by an orthogonal matrix (orthogonally diagonalizable). To orthogonally diagonalize A, one must first find its eigenvalues, and then find an orthonormal eigenbasis.