Symmetric Real-Number Matrix

A Symmetric Real-Number Matrix is a square real-number matrix (with real-numbers) that is a symmetric matrix.

• Context:
  • It can range from being a Symmetric Positive Real-Number Matrix to being a Symmetric Non-Negative Real-Number Matrix to being ...
• Example(s):
  • a Positive-Definite Matrix, such as [math]\displaystyle{ M = \begin{bmatrix} 2&-1&0\\-1&2&-1\\0&-1&2 \end{bmatrix} }[/math]
  • a Symmetric Binary Matrix.
  • a Symmetric Complex-Number Matrix.
  • …
• Counter-Example(s):
  • Asymmetric Real Matrix.
• See: Inner Product, Basis (Linear Algebra), Linear Operator, Differential Geometry, Tangent Space, Riemannian Manifold, Hilbert Space, Diagonal Matrix, Orthogonal Matrix.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/symmetric_matrix#Real_symmetric_matrices Retrieved:2015-1-19.
  • Denote by [math]\displaystyle{ \langle \cdot,\cdot \rangle }[/math] the standard inner product on Rⁿ. The real n-by-n matrix A is symmetric if and only if

     :[math]\displaystyle{ \langle Ax,y \rangle = \langle x, Ay\rangle \quad \forall x,y\in\Bbb{R}^n. }[/math]

    Since this definition is independent of the choice of 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 spaces.

    The finite-dimensional spectral theorem says that any symmetric matrix whose entries are real can be diagonalized by an orthogonal matrix. More explicitly: For every symmetric real matrix A there exists a real orthogonal matrix Q such that D = Q^TAQ is a diagonal matrix. Every symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix.

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

    Every real symmetric matrix is Hermitian, and therefore all its eigenvalues are real. (In fact, the eigenvalues are the entries in the diagonal matrix D (above), and therefore D is uniquely determined by A up to the order of its entries.) Essentially, the property of being symmetric for real matrices corresponds to the property of being Hermitian for complex matrices.

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