Wikipedia / Symmetric matrix
Wikipedia / Symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
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 diagonalth column then
Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from , 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.
Example
.
Properties
Basic properties
The sum and difference of two symmetric matrices are symmetric.
is also symmetric.
is symmetric.
.
Decomposition into symmetric and skew-symmetric
skew-symmetric matrices, then
i.e.
denotes the direct sum, then
. This is true for every square matrix with entries from any field whose characteristic is different from .
scalars (the number of entries on or above the main diagonal). Similarly, a skew-symmetric matrix scalars (the number of entries above the main diagonal).
Matrix congruent to a symmetric matrix
Any matrix congruent.
Symmetry and real values imply normality
A symmetric real matrix is necessarily a normal matrix.
Real symmetric matrices
the standard inner product
Since this definition is independent of the choice of basis, symmetry is a property that depends only on the linear operator and the choice of an 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 that of Hilbert spaces.
The finite-dimensional spectral theorem says that any symmetric matrix whose entries are real can be diagonalized by an orthogonal matrix is a diagonal matrix. Every real symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix.
real symmetric matrices that commute, then they can be simultaneously diagonalized by an orthogonal matrix: such that every element of the basis is an eigenvector.
Every real symmetric matrix is Hermitian, and therefore all its eigenvalues up to the order of its entries.) Essentially, the property of being symmetric for a real matrix corresponds to the property of being Hermitian for a complex matrix.
Complex symmetric matrices Complex
A complex symmetric matrix can be 'diagonalized' using a unitary matrix is a real diagonal matrix with non-negative entries. This result is referred to as the Autonne–Takagi factorization. It was originally proved by Léon Autonne (1915) and Teiji Takagi (1925), and rediscovered with different proofs by several other mathematicians.
is Hermitian and positive semi-definite, they coincide with the singular values may not be diagonalized by any similarity transformation.)
Decomposition
Using the Jordan normal form, one can prove that every square real matrix can be written as a product of two real symmetric matrices, and every square complex matrix can be written as a product of two complex symmetric matrices.
Every real non-singular matrix can be uniquely factored as the product of an orthogonal matrix and a symmetric positive definite matrix, which is called a polar decomposition. Singular matrices can also be factored, but not uniquely.
is a permutation matrix (arising from the need to pivot blocks, which is called Bunch–Kaufman decomposition
A general (complex) symmetric matrix may be defective and thus not be diagonalizable
.
Hessian
matrices of real functions appear as the Hessians real variables (the continuity of the second derivative is not needed, despite common belief to the opposite).
Every quadratic form which are generalizations of conic sections.
This is important partly because the second-order behavior of every smooth multi-variable function is described by the quadratic form belonging to the function's Hessian; this is a consequence of Taylor's theorem.
Symmetrizable matrix
is said to be symmetrizable if there exist an invertible diagonal matrix
is symmetrizable if and only if the following conditions are satisfied:
See also
Other types of symmetry or pattern in square matrices have special names; see for example:
Skew-symmetric matrix (also called antisymmetric or antimetric)
See also symmetry in mathematics.
Notes
References
External links