Symmetric matrix


In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,

Symmetry of a 5×5 matrix

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 diagonal. So if denotes the entry in the th row and th column then

for all indices and

Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, 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[1] 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 edit

The following   matrix is symmetric:

Since  .

Properties edit

Basic properties edit

  • The sum and difference of two symmetric matrices is symmetric.
  • This is not always true for the product: given symmetric matrices   and  , then   is symmetric if and only if   and   commute, i.e., if  .
  • For any integer  ,   is symmetric if   is symmetric.
  • If   exists, it is symmetric if and only if   is symmetric.
  • Rank of a symmetric matrix   is equal to the number of non-zero eigenvalues of  .

Decomposition into symmetric and skew-symmetric edit

Any square matrix can uniquely be written as sum of a symmetric and a skew-symmetric matrix. This decomposition is known as the Toeplitz decomposition. Let   denote the space of   matrices. If   denotes the space of   symmetric matrices and   the space of   skew-symmetric matrices then   and  , i.e.

where   denotes the direct sum. Let   then

Notice that   and  . This is true for every square matrix   with entries from any field whose characteristic is different from 2.

A symmetric   matrix is determined by   scalars (the number of entries on or above the main diagonal). Similarly, a skew-symmetric matrix is determined by   scalars (the number of entries above the main diagonal).

Matrix congruent to a symmetric matrix edit

Any matrix congruent to a symmetric matrix is again symmetric: if   is a symmetric matrix, then so is   for any matrix  .

Symmetry implies normality edit

A (real-valued) symmetric matrix is necessarily a normal matrix.

Real symmetric matrices edit

Denote by   the standard inner product on  . The real   matrix   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.

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 real symmetric matrix   there exists a real orthogonal matrix   such that   is a diagonal matrix. 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 Hermitian, and therefore all its eigenvalues are real. (In fact, the eigenvalues are the entries in the diagonal matrix   (above), and therefore   is uniquely determined by   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.

Complex symmetric matrices edit

A complex symmetric matrix can be 'diagonalized' using a unitary matrix: thus if   is a complex symmetric matrix, there is a unitary matrix   such that   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.[3][4] In fact, the matrix   is Hermitian and positive semi-definite, so there is a unitary matrix   such that   is diagonal with non-negative real entries. Thus   is complex symmetric with   real. Writing   with   and   real symmetric matrices,  . Thus  . Since   and   commute, there is a real orthogonal matrix   such that both   and   are diagonal. Setting   (a unitary matrix), the matrix   is complex diagonal. Pre-multiplying   by a suitable diagonal unitary matrix (which preserves unitarity of  ), the diagonal entries of   can be made to be real and non-negative as desired. To construct this matrix, we express the diagonal matrix as  . The matrix we seek is simply given by  . Clearly   as desired, so we make the modification  . Since their squares are the eigenvalues of  , they coincide with the singular values of  . (Note, about the eigen-decomposition of a complex symmetric matrix  , the Jordan normal form of   may not be diagonal, therefore   may not be diagonalized by any similarity transformation.)

Decomposition edit

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.[5]

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.

Cholesky decomposition states that every real positive-definite symmetric matrix   is a product of a lower-triangular matrix   and its transpose,


If the matrix is symmetric indefinite, it may be still decomposed as   where   is a permutation matrix (arising from the need to pivot),   a lower unit triangular matrix, and  [relevant?] is a direct sum of symmetric   and   blocks, which is called Bunch–Kaufman decomposition [6]

A general (complex) symmetric matrix may be defective and thus not be diagonalizable. If   is diagonalizable it may be decomposed as

where   is an orthogonal matrix  , and   is a diagonal matrix of the eigenvalues of  . In the special case that   is real symmetric, then   and   are also real. To see orthogonality, suppose   and   are eigenvectors corresponding to distinct eigenvalues  ,  . Then

Since   and   are distinct, we have  .

Hessian edit

Symmetric   matrices of real functions appear as the Hessians of twice differentiable functions of   real variables (the continuity of the second derivative is not needed, despite common belief to the opposite[7]).

Every quadratic form   on   can be uniquely written in the form   with a symmetric   matrix  . Because of the above spectral theorem, one can then say that every quadratic form, up to the choice of an orthonormal basis of  , "looks like"

with real numbers  . This considerably simplifies the study of quadratic forms, as well as the study of the level sets   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 edit

An   matrix   is said to be symmetrizable if there exists an invertible diagonal matrix   and symmetric matrix   such that  

The transpose of a symmetrizable matrix is symmetrizable, since   and   is symmetric. A matrix   is symmetrizable if and only if the following conditions are met:

  1.   implies   for all  
  2.   for any finite sequence  

See also edit

Other types of symmetry or pattern in square matrices have special names; see for example:

See also symmetry in mathematics.

Notes edit

  1. ^ Jesús Rojo García (1986). Álgebra lineal (in Spanish) (2nd ed.). Editorial AC. ISBN 84-7288-120-2.
  2. ^ Richard Bellman (1997). Introduction to Matrix Analysis (2nd ed.). SIAM. ISBN 08-9871-399-4.
  3. ^ Horn, R.A.; Johnson, C.R. (2013). Matrix analysis (2nd ed.). Cambridge University Press. pp. 263, 278. MR 2978290.
  4. ^ See:
    • Autonne, L. (1915), "Sur les matrices hypohermitiennes et sur les matrices unitaires", Ann. Univ. Lyon, 38: 1–77
    • Takagi, T. (1925), "On an algebraic problem related to an analytic theorem of Carathéodory and Fejér and on an allied theorem of Landau", Jpn. J. Math., 1: 83–93, doi:10.4099/jjm1924.1.0_83
    • Siegel, Carl Ludwig (1943), "Symplectic Geometry", American Journal of Mathematics, 65 (1): 1–86, doi:10.2307/2371774, JSTOR 2371774, Lemma 1, page 12
    • Hua, L.-K. (1944), "On the theory of automorphic functions of a matrix variable I–geometric basis", Amer. J. Math., 66 (3): 470–488, doi:10.2307/2371910, JSTOR 2371910
    • Schur, I. (1945), "Ein Satz über quadratische Formen mit komplexen Koeffizienten", Amer. J. Math., 67 (4): 472–480, doi:10.2307/2371974, JSTOR 2371974
    • Benedetti, R.; Cragnolini, P. (1984), "On simultaneous diagonalization of one Hermitian and one symmetric form", Linear Algebra Appl., 57: 215–226, doi:10.1016/0024-3795(84)90189-7
  5. ^ Bosch, A. J. (1986). "The factorization of a square matrix into two symmetric matrices". American Mathematical Monthly. 93 (6): 462–464. doi:10.2307/2323471. JSTOR 2323471.
  6. ^ G.H. Golub, C.F. van Loan. (1996). Matrix Computations. The Johns Hopkins University Press, Baltimore, London.
  7. ^ Dieudonné, Jean A. (1969). Foundations of Modern Analysis (Enlarged and Corrected printing ed.). Academic Press. pp. Theorem (8.12.2), p. 180. ISBN 978-1443724265.

References edit

  • Horn, Roger A.; Johnson, Charles R. (2013), Matrix analysis (2nd ed.), Cambridge University Press, ISBN 978-0-521-54823-6

External links edit

  • "Symmetric matrix", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  • A brief introduction and proof of eigenvalue properties of the real symmetric matrix
  • How to implement a Symmetric Matrix in C++