Symmetric matrix: Difference between revisions

Revision as of 04:35, 20 August 2021

Revision as of 04:07, 7 September 2021

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Since this definition is independent of the choice of [[basis (linear algebra)|basis]], symmetry is a property that depends only on the [[linear operator]] A and a choice of [[inner product]]. This characterization of symmetry is useful, for example, in [[differential geometry]], for each [[tangent space]] to a [[manifold]] may be endowed with an inner product, giving rise to what is called a [[Riemannian manifold]]. Another area where this formulation is used is in [[Hilbert space]]s.

The finite-dimensional [[spectral theorem]] says that any symmetric matrix whose entries are [[real number|real]] can be [[diagonal matrix|diagonalized]] by an [[orthogonal matrix]]. More explicitly: For every symmetric real symmetric matrix <math>A</math> there exists a real orthogonal matrix <math>Q</math> such that <math>D = Q^{\mathrm T} A Q</math> is a [[diagonal matrix]]. Every real symmetric matrix is thus, [[up to]] choice of an [[orthonormal basis]], a diagonal matrix.

If <math>A</math> and <math>B</math> are <math>n \times n</math> real symmetric matrices that commute, then they can be simultaneously diagonalized: there exists a basis of <math>\mathbb{R}^n</math> such that every element of the basis is an [[eigenvector]] for both <math>A</math> and <math>B</math>.