Cholesky decomposition


In linear algebra, the Cholesky decomposition or Cholesky factorization is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations. It was discovered by André-Louis Cholesky for real matrices. When it is applicable, the Cholesky decomposition is roughly twice as efficient as the LU decomposition for solving systems of linear equations.

Statement

The Cholesky decomposition of a Hermitian positive-definite matrix A, is a decomposition of the form
where L is a lower triangular matrix with real and positive diagonal entries, and L* denotes the conjugate transpose of L. Every Hermitian positive-definite matrix has a unique Cholesky decomposition.
The converse holds trivially: if A can be written as LL* for some invertible L, lower triangular or otherwise, then A is Hermitian and positive definite.
When A is symmetric, the factorization may be written
If A is real, so is L.

Positive semidefinite matrices

If a Hermitian matrix A is only positive semidefinite, instead of positive definite, then it still has a decomposition of the form A = LL* where the diagonal entries of L are allowed to be zero.
The decomposition need not be unique, for example:
However, if the rank of A is r, then there is a unique lower triangular L with exactly r positive diagonal elements and n-r columns containing all zeroes.
Alternatively, the decomposition can be made unique when a pivoting choice is fixed. Formally, if A is an n × n positive semidefinite matrix of rank r, then there is at least one permutation matrix P such that P A PT has a unique decomposition of the form P A PT L L* with
where L1 is an r × r lower triangular matrix with positive diagonal.

LDL decomposition

A closely related variant of the classical Cholesky decomposition is the LDL decomposition,
where L is a lower unit triangular matrix, and D is a diagonal matrix.
That is, the diagonal elements of L are required to be 1 at the cost of introducing an additional diagonal matrix D in the decomposition.
The main advantage is that the LDL decomposition can be computed and used with essentially the same algorithms, but avoids extracting square roots.
For this reason, the LDL decomposition is often called the square-root-free Cholesky decomposition. For real matrices, the factorization has the form A = LDLT and is often referred to as LDLT decomposition. It is closely related to the eigendecomposition of real symmetric matrices, A = QΛQT.
The LDL decomposition is related to the classical Cholesky decomposition of the form LL* as follows:
Conversely, given the classical Cholesky decomposition of a positive definite matrix, if S is a diagonal matrix that contains the main diagonal of, then a A can be decomposed as where
If A is positive definite then the diagonal elements of D are all positive.
For positive semidefinite A, an decomposition exists where the number of non-zero elements on the diagonal D is exactly the rank of A.
Some indefinite matrices for which no Cholesky decomposition exists have an LDL decomposition with negative entries in D: it suffices that the first n-1 leading principal minors of A are non-singular.

Example

Here is the Cholesky decomposition of a symmetric real matrix:
And here is its LDLT decomposition:

Applications

The Cholesky decomposition is mainly used for the numerical solution of linear equations. If A is symmetric and positive definite, then we can solve by first computing the Cholesky decomposition , then solving for y by forward substitution, and finally solving for x by back substitution.
An alternative way to eliminate taking square roots in the decomposition is to compute the Cholesky decomposition, then solving for y, and finally solving.
For linear systems that can be put into symmetric form, the Cholesky decomposition is the method of choice, for superior efficiency and numerical stability. Compared to the LU decomposition, it is roughly twice as efficient.

Linear least squares

Systems of the form Ax = b with A symmetric and positive definite arise quite often in applications. For instance, the normal equations in linear least squares problems are of this form. It may also happen that matrix A comes from an energy functional, which must be positive from physical considerations; this happens frequently in the numerical solution of partial differential equations.

Non-linear optimization

Non-linear multi-variate functions may be minimized over their parameters using variants of Newton's method called quasi-Newton methods. At iteration k, the search steps in a direction defined by solving = for, where is the step direction, is the gradient, and is an approximation to the Hessian matrix formed by repeating rank-1 updates at each iteration. Two well-known update formulas are called Davidon–Fletcher–Powell and Broyden–Fletcher–Goldfarb–Shanno. Loss of the positive-definite condition through round-off error is avoided if rather than updating an approximation to the inverse of the Hessian, one updates the Cholesky decomposition of an approximation of the Hessian matrix itself

Monte Carlo simulation

The Cholesky decomposition is commonly used in the Monte Carlo method for simulating systems with multiple correlated variables. The covariance matrix is decomposed to give the lower-triangular L. Applying this to a vector of uncorrelated samples u produces a sample vector Lu with the covariance properties of the system being modeled.
The following simplified example shows the economy one gets from the Cholesky decomposition: suppose the goal is to generate two correlated normal variables and with given correlation coefficient. To accomplish that, it is necessary to first generate two uncorrelated Gaussian random variables and, which can be done using a Box–Muller transform. Given the required correlation coefficient, the correlated normal variables can be obtained via the transformations and.

Kalman filters

s commonly use the Cholesky decomposition to choose a set of so-called sigma points. The Kalman filter tracks the average state of a system as a vector x of length N and covariance as an N × N matrix P. The matrix P is always positive semi-definite and can be decomposed into LLT. The columns of L can be added and subtracted from the mean x to form a set of 2N vectors called sigma points. These sigma points completely capture the mean and covariance of the system state.

Matrix inversion

The explicit inverse of a Hermitian matrix can be computed by Cholesky decomposition, in a manner similar to solving linear systems, using operations. The entire inversion can even be efficiently performed in-place.
A non-Hermitian matrix B can also be inverted using the following identity, where BB* will always be Hermitian:

Computation

There are various methods for calculating the Cholesky decomposition. The computational complexity of commonly used algorithms is O in general. The algorithms described below all involve about n3/3 FLOPs, where n is the size of the matrix A. Hence, they have half the cost of the LU decomposition, which uses 2n3/3 FLOPs.
Which of the algorithms below is faster depends on the details of the implementation. Generally, the first algorithm will be slightly slower because it accesses the data in a less regular manner.

The Cholesky algorithm

The Cholesky algorithm, used to calculate the decomposition matrix L, is a modified version of Gaussian elimination.
The recursive algorithm starts with i := 1 and
At step i, the matrix A has the following form:
where Ii−1 denotes the identity matrix of dimension i − 1.
If we now define the matrix Li by
then we can write A as
where
Note that bi b*i is an outer product, therefore this algorithm is called the outer-product version in.
We repeat this for i from 1 to n. After n steps, we get A = I. Hence, the lower triangular matrix L we are looking for is calculated as

The Cholesky–Banachiewicz and Cholesky–Crout algorithms

If we write out the equation
we obtain the following:
and therefore the following formulas for the entries of L:
For complex and real matrices, inconsequential arbitrary sign changes of diagonal and associated off-diagonal elements are allowed. The expression under the square root is always positive if A is real and positive-definite.
For complex Hermitian matrix, the following formula applies:
So we can compute the entry if we know the entries to the left and above. The computation is usually arranged in either of the following orders:
Either pattern of access allows the entire computation to be performed in-place if desired.

Stability of the computation

Suppose that we want to solve a well-conditioned system of linear equations. If the LU decomposition is used, then the algorithm is unstable unless we use some sort of pivoting strategy. In the latter case, the error depends on the so-called growth factor of the matrix, which is usually small.
Now, suppose that the Cholesky decomposition is applicable. As mentioned above, the algorithm will be twice as fast. Furthermore, no pivoting is necessary, and the error will always be small. Specifically, if we want to solve Ax = b, and y denotes the computed solution, then y solves the perturbed system y = b, where
Here ||·||2 is the matrix 2-norm, cn is a small constant depending on n, and ε denotes the unit round-off.
One concern with the Cholesky decomposition to be aware of is the use of square roots. If the matrix being factorized is positive definite as required, the numbers under the square roots are always positive in exact arithmetic. Unfortunately, the numbers can become negative because of round-off errors, in which case the algorithm cannot continue. However, this can only happen if the matrix is very ill-conditioned. One way to address this is to add a diagonal correction matrix to the matrix being decomposed in an attempt to promote the positive-definiteness. While this might lessen the accuracy of the decomposition, it can be very favorable for other reasons; for example, when performing Newton's method in optimization, adding a diagonal matrix can improve stability when far from the optimum.

LDL decomposition

An alternative form, eliminating the need to take square roots when A is symmetric, is the symmetric indefinite factorization
The following recursive relations apply for the entries of D and L:
This works as long as the generated diagonal elements in D stay non-zero. The decomposition is then unique. D and L are real if A is real.
For complex Hermitian matrix A, the following formula applies:
Again, the pattern of access allows the entire computation to be performed in-place if desired.

Block variant

When used on indefinite matrices, the LDL* factorization is known to be unstable without careful pivoting; specifically, the elements of the factorization can grow arbitrarily. A possible improvement is to perform the factorization on block sub-matrices, commonly 2 × 2:
where every element in the matrices above is a square submatrix. From this, these analogous recursive relations follow:
This involves matrix products and explicit inversion, thus limiting the practical block size.

Updating the decomposition

A task that often arises in practice is that one needs to update a Cholesky decomposition. In more details, one has already computed the Cholesky decomposition of some matrix, then one changes the matrix in some way into another matrix, say, and one wants to compute the Cholesky decomposition of the updated matrix:. The question is now whether one can use the Cholesky decomposition of that was computed before to compute the Cholesky decomposition of.

Rank-one update

The specific case, where the updated matrix is related to the matrix by, is known as a rank-one update.
Here is a little function written in Matlab syntax that realizes a rank-one update:

function = cholupdate
n = length;
for k = 1:n
r = sqrt^2 + x;
c = r / L;
s = x / L;
L = r;
if k < n
L = + s * x) / c;
x = c * x - s * L;
end
end
end

Rank-one downdate

A rank-one downdate is similar to a rank-one update, except that the addition is replaced by subtraction:. This only works if the new matrix is still positive definite.
The code for the rank-one update shown above can easily be adapted to do a rank-one downdate: one merely needs to replace the two additions in the assignment to r and L by subtractions.

Adding and removing rows and columns

If we have a symmetric and positive definite matrix represented in block form as
and its upper Cholesky factor
then for a new matrix, which is the same as but with the insertion of new rows and columns,
we are interested in finding the Cholesky factorisation of, which we call, without directly computing the entire decomposition.
Writing for the solution of, which can be found easily for triangular matrices, and for the Cholesky decomposition of, the following relations can be found:
These formulas may be used to determine the Cholesky factor after the insertion of rows or columns in any position, if we set the row and column dimensions appropriately. The inverse problem, when we have
with known Cholesky decomposition
and wish to determine the Cholesky factor
of the matrix with rows and columns removed,
yields the following rules:
Notice that the equations above that involve finding the Cholesky decomposition of a new matrix are all of the form, which allows them to be efficiently calculated using the update and downdate procedures detailed in the previous section.

Proof for positive semi-definite matrices

Proof by limiting argument

The above algorithms show that every positive definite matrix has a Cholesky decomposition. This result can be extended to the positive semi-definite case by a limiting argument. The argument is not fully constructive, i.e., it gives no explicit numerical algorithms for computing Cholesky factors.
If is an positive semi-definite matrix, then the sequence consists of positive definite matrices. Also,
in operator norm. From the positive definite case, each has Cholesky decomposition. By property of the operator norm,
So is a bounded set in the Banach space of operators, therefore relatively compact.
Consequently, it has a convergent subsequence, also denoted by, with limit.
It can be easily checked that this has the desired properties, i.e., and is lower triangular with non-negative diagonal entries: for all and,
Therefore,.
Because the underlying vector space is finite-dimensional, all topologies on the space of operators are equivalent.
So tends to in norm means tends to entrywise.
This in turn implies that, since each is lower triangular with non-negative diagonal entries, is also.

Proof by QR decomposition

Let be a positive semi-definite matrix Hermitian matrix. Then it can be written as a product of its square root matrix,. Now QR decomposition can be applied to, resulting in
, where is unitary and is upper triangular. Inserting the decomposition into the original equality yields. Setting completes the proof.

Generalization

The Cholesky factorization can be generalized to matrices with operator entries. Let be a sequence of Hilbert spaces. Consider the operator matrix
acting on the direct sum
where each
is a bounded operator. If A is positive in the sense that for all finite k and for any
we have, then there exists a lower triangular operator matrix L such that A = LL*. One can also take the diagonal entries of L to be positive.

Implementations in programming libraries