Symmetric Matrix - Properties

Properties

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 = QTAQ is a diagonal matrix. Every symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix.

Another way to phrase the spectral theorem is that a real n×n matrix A is symmetric if and only if there is an orthonormal basis of consisting of eigenvectors for A.

Every real symmetric matrix is Hermitian, and therefore all its eigenvalues are real. (In fact, the eigenvalues are the entries in the above diagonal matrix D, 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.

A complex symmetric matrix A can often, but not always, be diagonalized in the form D = UT A U, where D is complex diagonal and U is not Hermitian but complex orthogonal with UT U = I. In this case the columns of U are the eigenvectors of A and the diagonal elements of D are eigenvalues. An example of a complex symmetric matrix that cannot be diagonalized is

The sum and difference of two symmetric matrices is again symmetric, but this is not always true for the product: given symmetric matrices A and B, then AB is symmetric if and only if A and B commute, i.e., if AB = BA. So for integer n, An is symmetric if A is symmetric. If A and B are n×n real symmetric matrices that commute, then there exists a basis of such that every element of the basis is an eigenvector for both A and B.

If A−1 exists, it is symmetric if and only if A is symmetric.

Let Mat_n denote the space of n × n matrices. A symmetric n × n matrix is determined by n(n + 1)/2 scalars (the number of entries on or above the main diagonal). Similarly, a skew-symmetric matrix is determined by n(n − 1)/2 scalars (the number of entries above the main diagonal). If Sym_n denotes the space of n × n symmetric matrices and Skew_n the space of n × n skew-symmetric matrices then since Mat_n = Sym_n + Skew_n and Sym_n ∩ Skew_n = {0}, i.e.

where ⊕ denotes the direct sum. Let X ∈ Mat_n then

Notice that ½(X + XT) ∈ Sym_n and ½(X − XT) ∈ Skew_n. This is true for every square matrix X with entries from any field whose characteristic is different from 2.

Any matrix congruent to a symmetric matrix is again symmetric: if X is a symmetric matrix then so is AXAT for any matrix A.

Denote with the standard inner product on Rn. The real n-by-n matrix A is symmetric if and only if

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.

A symmetric matrix is a normal matrix.