mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Weinstein–Aronszajn identity states that if

A

and

B

are

matrices 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 ...

of size and respectively (either or both of which may be infinite) then, provided

AB

(and hence, also

BA

) is of

trace class In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace ...

, :

\det(I_m + AB) = \det(I_n + BA),

where

I_k

is the

identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...

. It is closely related to the

matrix determinant lemma In mathematics, in particular linear algebra, the matrix determinant lemma computes the determinant of the sum of an invertible matrix A and the dyadic product, uvT, of a column vector u and a row vector vT. Statement Suppose A is an invertib ...

and its generalization. It is the

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...

analogue of the

Woodbury matrix identity In mathematics (specifically linear algebra), the Woodbury matrix identity, named after Max A. Woodbury, says that the inverse of a rank-''k'' correction of some matrix can be computed by doing a rank-''k'' correction to the inverse of the origina ...

for matrix inverses.

Proof

The identity may be proved as follows. Let

M

be a matrix comprising the four blocks

I_m

-A

B

and

I_n

. :

M = \begin I_m & -A \\ B & I_n \end.

Because is

invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...

, the formula for the determinant of a block matrix gives :

\det\begin I_m & -A \\ B & I_n \end = \det(I_m) \det\left(I_n - B I_m^ (-A)\right) = \det(I_n + BA).

Because is invertible, the formula for the determinant of a block matrix gives :

\det\begin I_m & -A\\ B & I_n \end = \det(I_n) \det\left(I_m - (-A) I_n^ B\right) = \det(I_m + AB).

Thus :

\det(I_n + B A) = \det(I_m + A B).

Applications

Let

\lambda \in \mathbb \setminus \

. The identity can be used to show the somewhat more general statement that :

\det(AB - \lambda I_m) = (-\lambda)^ \det(BA - \lambda I_n).

It follows that the non-zero

eigenvalues 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 denote ...

AB

and

BA

are the same. This identity is useful in developing a

Bayes estimator In estimation theory and decision theory, a Bayes estimator or a Bayes action is an estimator or decision rule that minimizes the posterior expected value of a loss function (i.e., the posterior expected loss). Equivalently, it maximizes the ...

for

multivariate Gaussian distribution In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One d ...

s. The identity also finds applications in

random matrix theory In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathemat ...

by relating determinants of large matrices to determinants of smaller ones.

References

Determinants Matrix theory Theorems in linear algebra {{linear-algebra-stub