In
matrix theory
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.
For example,
\beg ...
, Sylvester's formula or Sylvester's matrix theorem (named after
J. J. Sylvester) or Lagrange−Sylvester interpolation expresses an analytic
function of a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
as a polynomial in , in terms of the
eigenvalues and eigenvectors
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
of .
[/
Roger A. Horn and Charles R. Johnson (1991), ''Topics in Matrix Analysis''. Cambridge University Press,
][ Jon F. Claerbout (1976), ''Sylvester's matrix theorem'', a section of ''Fundamentals of Geophysical Data Processing'']
Online version
at sepwww.stanford.edu, accessed on 2010-03-14.
It states that
:
where the are the eigenvalues of , and the matrices
:
are the corresponding
Frobenius covariant In matrix theory, the Frobenius covariants of a square matrix are special polynomials of it, namely projection matrices ''A'i'' associated with the eigenvalues and eigenvectors of .Roger A. Horn and Charles R. Johnson (1991), ''Topics in M ...
s of , which are (projection) matrix
Lagrange polynomials
In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data.
Given a data set of coordinate pairs (x_j, y_j) with 0 \leq j \leq k, the x_j are called ''nodes'' ...
of .
Conditions
Sylvester's formula applies for any
diagonalizable matrix
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) ...
with distinct eigenvalues,
1, …, ''λ''
''k'', and any function defined on some subset of the
complex numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
such that is well defined. The last condition means that every eigenvalue is in the domain of , and that every eigenvalue with multiplicity
''i'' > 1 is in the interior of the domain, with being () times differentiable at .
[
]
Example
Consider the two-by-two matrix:
:
This matrix has two eigenvalues, 5 and −2. Its Frobenius covariants are
:
Sylvester's formula then amounts to
:
For instance, if is defined by , then Sylvester's formula expresses the matrix inverse as
:
Generalization
Sylvester's formula is only valid for diagonalizable matrices
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
; an extension due to Arthur Buchheim, based on Hermite interpolating polynomials, covers the general case:
: