Cholesky decomposition

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.[7] The decomposition need not be unique, for example:

${\displaystyle {\begin{bmatrix}0&0\\0&1\end{bmatrix}}=\mathbf {L} \mathbf {L} ^{*},\quad \quad \mathbf {L} ={\begin{bmatrix}0&0\\\cos \theta &\sin \theta \end{bmatrix}}.}$

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.[8]

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 P^T has a unique decomposition of the form P A P^T = L L^* with ${\displaystyle \mathbf {L} ={\begin{bmatrix}\mathbf {L} _{1}&0\\\mathbf {L} _{2}&0\end{bmatrix}}}$, where L₁ is an r × r lower triangular matrix with positive diagonal.[9]

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 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 ${\displaystyle p_{k}}$ defined by solving ${\displaystyle B_{k}p_{k}=-g_{k}}$ for ${\displaystyle p_{k}}$, where ${\displaystyle p_{k}}$ is the step direction, ${\displaystyle g_{k}}$ is the gradient, and ${\displaystyle B_{k}}$ 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 (DFP) and Broyden–Fletcher–Goldfarb–Shanno (BFGS). 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 .[14]

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.[15]

The following simplified example shows the economy one gets from the Cholesky decomposition: suppose the goal is to generate two correlated normal variables ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$ with given correlation coefficient ${\displaystyle \rho }$. To accomplish that, it is necessary to first generate two uncorrelated Gaussian random variables ${\displaystyle z_{1}}$ and ${\displaystyle z_{2}}$, which can be done using a Box–Muller transform. Given the required correlation coefficient ${\displaystyle \rho }$, the correlated normal variables can be obtained via the transformations ${\displaystyle x_{1}=z_{1}}$ and ${\textstyle x_{2}=\rho z_{1}+{\sqrt {1-\rho ^{2}}}z_{2}}$.

Unscented Kalman filters 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 LL^T. 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.

The explicit inverse of a Hermitian matrix can be computed by Cholesky decomposition, in a manner similar to solving linear systems, using ${\displaystyle n^{3}}$ operations (${\displaystyle {\tfrac {1}{2}}n^{3}}$ multiplications).[10] 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:

${\displaystyle \mathbf {B} ^{-1}=\mathbf {B} ^{*}(\mathbf {BB} ^{*})^{-1}.}$

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

A⁽¹⁾ := A.

At step i, the matrix A⁽ⁱ⁾ has the following form:

${\displaystyle \mathbf {A} ^{(i)}={\begin{pmatrix}\mathbf {I} _{i-1}&0&0\\0&a_{i,i}&\mathbf {b} _{i}^{*}\\0&\mathbf {b} _{i}&\mathbf {B} ^{(i)}\end{pmatrix}},}$

where I_i−1 denotes the identity matrix of dimension i − 1.

If the matrix L_i is defined by

${\displaystyle \mathbf {L} _{i}:={\begin{pmatrix}\mathbf {I} _{i-1}&0&0\\0&{\sqrt {a_{i,i}}}&0\\0&{\frac {1}{\sqrt {a_{i,i}}}}\mathbf {b} _{i}&\mathbf {I} _{n-i}\end{pmatrix}},}$

(note that a_i,i > 0 since A⁽ⁱ⁾ is positive definite), then A⁽ⁱ⁾ can be written as

${\displaystyle \mathbf {A} ^{(i)}=\mathbf {L} _{i}\mathbf {A} ^{(i+1)}\mathbf {L} _{i}^{*}}$

where

${\displaystyle \mathbf {A} ^{(i+1)}={\begin{pmatrix}\mathbf {I} _{i-1}&0&0\\0&1&0\\0&0&\mathbf {B} ^{(i)}-{\frac {1}{a_{i,i}}}\mathbf {b} _{i}\mathbf {b} _{i}^{*}\end{pmatrix}}.}$

Note that b_i b*_i is an outer product, therefore this algorithm is called the outer-product version in (Golub & Van Loan).

This is repeated for i from 1 to n. After n steps, A⁽ⁿ⁺¹⁾ = I is obtained, and hence, the lower triangular matrix L sought for is calculated as

${\displaystyle \mathbf {L}$ :=\mathbf {L} _{1}\mathbf {L} _{2}\dots \mathbf {L} _{n}.}

Access pattern (white) and writing pattern (yellow) for the in-place Cholesky—Banachiewicz algorithm on a 5×5 matrix

If the equation

${\displaystyle {\begin{aligned}\mathbf {A} =\mathbf {LL} ^{T}&={\begin{pmatrix}L_{11}&0&0\\L_{21}&L_{22}&0\\L_{31}&L_{32}&L_{33}\\\end{pmatrix}}{\begin{pmatrix}L_{11}&L_{21}&L_{31}\\0&L_{22}&L_{32}\\0&0&L_{33}\end{pmatrix}}\\[8pt]&={\begin{pmatrix}L_{11}^{2}&&({\text{symmetric}})\\L_{21}L_{11}&L_{21}^{2}+L_{22}^{2}&\\L_{31}L_{11}&L_{31}L_{21}+L_{32}L_{22}&L_{31}^{2}+L_{32}^{2}+L_{33}^{2}\end{pmatrix}},\end{aligned}}}$

is written out, the following is obtained:

${\displaystyle {\begin{aligned}\mathbf {L} ={\begin{pmatrix}{\sqrt {A_{11}}}&0&0\\A_{21}/L_{11}&{\sqrt {A_{22}-L_{21}^{2}}}&0\\A_{31}/L_{11}&\left(A_{32}-L_{31}L_{21}\right)/L_{22}&{\sqrt {A_{33}-L_{31}^{2}-L_{32}^{2}}}\end{pmatrix}}\end{aligned}}}$

and therefore the following formulas for the entries of L:

${\displaystyle L_{j,j}=(\pm ){\sqrt {A_{j,j}-\sum _{k=1}^{j-1}L_{j,k}^{2}}},}$

${\displaystyle L_{i,j}={\frac {1}{L_{j,j}}}\left(A_{i,j}-\sum _{k=1}^{j-1}L_{i,k}L_{j,k}\right)\quad {\text{for }}i>j.}$

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:

${\displaystyle L_{j,j}={\sqrt {A_{j,j}-\sum _{k=1}^{j-1}L_{j,k}^{*}L_{j,k}}},}$

${\displaystyle L_{i,j}={\frac {1}{L_{j,j}}}\left(A_{i,j}-\sum _{k=1}^{j-1}L_{j,k}^{*}L_{i,k}\right)\quad {\text{for }}i>j.}$

So it now is possible to compute the (i, j) entry if the entries to the left and above are known. The computation is usually arranged in either of the following orders:

• The Cholesky–Banachiewicz algorithm starts from the upper left corner of the matrix L and proceeds to calculate the matrix row by row.

for (i = 0; i < dimensionSize; i++) {
    for (j = 0; j <= i; j++) {
        float sum = 0;
        for (k = 0; k < j; k++)
            sum += L[i][k] * L[j][k];

        if (i == j)
            L[i][j] = sqrt(A[i][i] - sum);
        else
            L[i][j] = (1.0 / L[j][j] * (A[i][j] - sum));
    }
}

The above algorithm can be succinctly expressed as combining a dot product and matrix multiplication in vectorized programming languages such as Fortran as the following,

do i = 1, size(A,1)
    L(i,i) = sqrt(A(i,i) - dot_product(L(i,1:i-1), L(i,1:i-1)))
    L(i+1:,i) = (A(i+1:,i) - matmul(conjg(L(i,1:i-1)), L(i+1:,1:i-1))) / L(i,i)
end do

where conjg refers to complex conjugate of the elements.

• The Cholesky–Crout algorithm starts from the upper left corner of the matrix L and proceeds to calculate the matrix column by column.

  for (j = 0; j < dimensionSize; j++) {
      float sum = 0;
      for (k = 0; k < j; k++) {
          sum += L[j][k] * L[j][k];
      }
      L[j][j] = sqrt(A[j][j] - sum);
  
      for (i = j + 1; i < dimensionSize; i++) {
          sum = 0;
          for (k = 0; k < j; k++) {
              sum += L[i][k] * L[j][k];
          }
          L[i][j] = (1.0 / L[j][j] * (A[i][j] - sum));
      }
  }
  

The above algorithm can be succinctly expressed as combining a dot product and matrix multiplication in vectorized programming languages such as Fortran as the following,

do i = 1, size(A,1)
    L(i,i) = sqrt(A(i,i) - dot_product(L(1:i-1,i), L(1:i-1,i)))
    L(i,i+1:) = (A(i,i+1:) - matmul(conjg(L(1:i-1,i)), L(1:i-1,i+1:))) / L(i,i)
end do

where conjg refers to complex conjugate of the elements.

Either pattern of access allows the entire computation to be performed in-place if desired.

Suppose that there is a desire to solve a well-conditioned system of linear equations. If the LU decomposition is used, then the algorithm is unstable unless some sort of pivoting strategy is used. In the latter case, the error depends on the so-called growth factor of the matrix, which is usually (but not always) 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 Ax = b, and y denotes the computed solution, then y solves the perturbed system (A + E)y = b, where

${\displaystyle \|\mathbf {E} \|_{2}\leq c_{n}\varepsilon \|\mathbf {A} \|_{2}.}$

Here ||·||₂ is the matrix 2-norm, c_n 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.[17] 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.

An alternative form, eliminating the need to take square roots when A is symmetric, is the symmetric indefinite factorization[18]

${\displaystyle {\begin{aligned}\mathbf {A} =\mathbf {LDL} ^{\mathrm {T} }&={\begin{pmatrix}1&0&0\\L_{21}&1&0\\L_{31}&L_{32}&1\\\end{pmatrix}}{\begin{pmatrix}D_{1}&0&0\\0&D_{2}&0\\0&0&D_{3}\\\end{pmatrix}}{\begin{pmatrix}1&L_{21}&L_{31}\\0&1&L_{32}\\0&0&1\\\end{pmatrix}}\\[8pt]&={\begin{pmatrix}D_{1}&&(\mathrm {symmetric} )\\L_{21}D_{1}&L_{21}^{2}D_{1}+D_{2}&\\L_{31}D_{1}&L_{31}L_{21}D_{1}+L_{32}D_{2}&L_{31}^{2}D_{1}+L_{32}^{2}D_{2}+D_{3}.\end{pmatrix}}.\end{aligned}}}$

The following recursive relations apply for the entries of D and L:

${\displaystyle D_{j}=A_{jj}-\sum _{k=1}^{j-1}L_{jk}^{2}D_{k},}$

${\displaystyle L_{ij}={\frac {1}{D_{j}}}\left(A_{ij}-\sum _{k=1}^{j-1}L_{ik}L_{jk}D_{k}\right)\quad {\text{for }}i>j.}$

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:

${\displaystyle D_{j}=A_{jj}-\sum _{k=1}^{j-1}L_{jk}L_{jk}^{*}D_{k},}$

${\displaystyle L_{ij}={\frac {1}{D_{j}}}\left(A_{ij}-\sum _{k=1}^{j-1}L_{ik}L_{jk}^{*}D_{k}\right)\quad {\text{for }}i>j.}$

Again, the pattern of access allows the entire computation to be performed in-place if desired.

When used on indefinite matrices, the LDL* factorization is known to be unstable without careful pivoting;[19] specifically, the elements of the factorization can grow arbitrarily. A possible improvement is to perform the factorization on block sub-matrices, commonly 2 × 2:[20]

${\displaystyle {\begin{aligned}\mathbf {A} =\mathbf {LDL} ^{\mathrm {T} }&={\begin{pmatrix}\mathbf {I} &0&0\\\mathbf {L} _{21}&\mathbf {I} &0\\\mathbf {L} _{31}&\mathbf {L} _{32}&\mathbf {I} \\\end{pmatrix}}{\begin{pmatrix}\mathbf {D} _{1}&0&0\\0&\mathbf {D} _{2}&0\\0&0&\mathbf {D} _{3}\\\end{pmatrix}}{\begin{pmatrix}\mathbf {I} &\mathbf {L} _{21}^{\mathrm {T} }&\mathbf {L} _{31}^{\mathrm {T} }\\0&\mathbf {I} &\mathbf {L} _{32}^{\mathrm {T} }\\0&0&\mathbf {I} \\\end{pmatrix}}\\[8pt]&={\begin{pmatrix}\mathbf {D} _{1}&&(\mathrm {symmetric} )\\\mathbf {L} _{21}\mathbf {D} _{1}&\mathbf {L} _{21}\mathbf {D} _{1}\mathbf {L} _{21}^{\mathrm {T} }+\mathbf {D} _{2}&\\\mathbf {L} _{31}\mathbf {D} _{1}&\mathbf {L} _{31}\mathbf {D} _{1}\mathbf {L} _{21}^{\mathrm {T} }+\mathbf {L} _{32}\mathbf {D} _{2}&\mathbf {L} _{31}\mathbf {D} _{1}\mathbf {L} _{31}^{\mathrm {T} }+\mathbf {L} _{32}\mathbf {D} _{2}\mathbf {L} _{32}^{\mathrm {T} }+\mathbf {D} _{3}\end{pmatrix}},\end{aligned}}}$

where every element in the matrices above is a square submatrix. From this, these analogous recursive relations follow:

${\displaystyle \mathbf {D} _{j}=\mathbf {A} _{jj}-\sum _{k=1}^{j-1}\mathbf {L} _{jk}\mathbf {D} _{k}\mathbf {L} _{jk}^{\mathrm {T} },}$

${\displaystyle \mathbf {L} _{ij}=\left(\mathbf {A} _{ij}-\sum _{k=1}^{j-1}\mathbf {L} _{ik}\mathbf {D} _{k}\mathbf {L} _{jk}^{\mathrm {T} }\right)\mathbf {D} _{j}^{-1}.}$

This involves matrix products and explicit inversion, thus limiting the practical block size.

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 ${\displaystyle \mathbf {A} =\mathbf {L} \mathbf {L} ^{*}}$ of some matrix ${\displaystyle \mathbf {A} }$, then one changes the matrix ${\displaystyle \mathbf {A} }$ in some way into another matrix, say ${\displaystyle {\tilde {\mathbf {A} }}}$, and one wants to compute the Cholesky decomposition of the updated matrix: ${\displaystyle {\tilde {\mathbf {A} }}={\tilde {\mathbf {L} }}{\tilde {\mathbf {L} }}^{*}}$. The question is now whether one can use the Cholesky decomposition of ${\displaystyle \mathbf {A} }$ that was computed before to compute the Cholesky decomposition of ${\displaystyle {\tilde {\mathbf {A} }}}$.

The specific case, where the updated matrix ${\displaystyle {\tilde {\mathbf {A} }}}$ is related to the matrix ${\displaystyle \mathbf {A} }$ by ${\displaystyle {\tilde {\mathbf {A} }}=\mathbf {A} +\mathbf {x} \mathbf {x} ^{*}}$, is known as a rank-one update.

Here is a function[21] written in Matlab syntax that realizes a rank-one update:

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

A rank-n update is one where for a matrix ${\displaystyle \mathbf {M} }$ one updates the decomposition such that ${\displaystyle {\tilde {\mathbf {A} }}=\mathbf {A} +\mathbf {M} \mathbf {M} ^{*}}$. This can be achieved by successively performing rank-one updates for each of the columns of ${\displaystyle \mathbf {M} }$.

A rank-one downdate is similar to a rank-one update, except that the addition is replaced by subtraction: ${\displaystyle {\tilde {\mathbf {A} }}=\mathbf {A} -\mathbf {x} \mathbf {x} ^{*}}$. This only works if the new matrix ${\displaystyle {\tilde {\mathbf {A} }}}$ 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((k+1):n, k) by subtractions.

If a symmetric and positive definite matrix ${\displaystyle \mathbf {A} }$ is represented in block form as

${\displaystyle \mathbf {A} ={\begin{pmatrix}\mathbf {A} _{11}&\mathbf {A} _{13}\\\mathbf {A} _{13}^{\mathrm {T} }&\mathbf {A} _{33}\\\end{pmatrix}}}$

and its upper Cholesky factor

${\displaystyle \mathbf {L} ={\begin{pmatrix}\mathbf {L} _{11}&\mathbf {L} _{13}\\0&\mathbf {L} _{33}\\\end{pmatrix}},}$

then for a new matrix ${\displaystyle {\tilde {\mathbf {A} }}}$, which is the same as ${\displaystyle \mathbf {A} }$ but with the insertion of new rows and columns,

${\displaystyle {\begin{aligned}{\tilde {\mathbf {A} }}&={\begin{pmatrix}\mathbf {A} _{11}&\mathbf {A} _{12}&\mathbf {A} _{13}\\\mathbf {A} _{12}^{\mathrm {T} }&\mathbf {A} _{22}&\mathbf {A} _{23}\\\mathbf {A} _{13}^{\mathrm {T} }&\mathbf {A} _{23}^{\mathrm {T} }&\mathbf {A} _{33}\\\end{pmatrix}}\end{aligned}}}$

Now there is an interest in finding the Cholesky factorization of ${\displaystyle {\tilde {\mathbf {A} }}}$, which can be called ${\displaystyle {\tilde {\mathbf {S} }}}$, without directly computing the entire decomposition.

${\displaystyle {\begin{aligned}{\tilde {\mathbf {S} }}&={\begin{pmatrix}\mathbf {S} _{11}&\mathbf {S} _{12}&\mathbf {S} _{13}\\0&\mathbf {S} _{22}&\mathbf {S} _{23}\\0&0&\mathbf {S} _{33}\\\end{pmatrix}}.\end{aligned}}}$

Writing ${\displaystyle \mathbf {A} \setminus \mathbf {b} }$ for the solution of ${\displaystyle \mathbf {A} \mathbf {x} =\mathbf {b} }$, which can be found easily for triangular matrices, and ${\displaystyle {\text{chol}}(\mathbf {M} )}$ for the Cholesky decomposition of ${\displaystyle \mathbf {M} }$, the following relations can be found:

${\displaystyle {\begin{aligned}\mathbf {S} _{11}&=\mathbf {L} _{11},\\\mathbf {S} _{12}&=\mathbf {L} _{11}^{\mathrm {T} }\setminus \mathbf {A} _{12},\\\mathbf {S} _{13}&=\mathbf {L} _{13},\\\mathbf {S} _{22}&=\mathrm {chol} \left(\mathbf {A} _{22}-\mathbf {S} _{12}^{\mathrm {T} }\mathbf {S} _{12}\right),\\\mathbf {S} _{23}&=\mathbf {S} _{22}^{\mathrm {T} }\setminus \left(\mathbf {A} _{23}-\mathbf {S} _{12}^{\mathrm {T} }\mathbf {S} _{13}\right),\\\mathbf {S} _{33}&=\mathrm {chol} \left(\mathbf {L} _{33}^{\mathrm {T} }\mathbf {L} _{33}-\mathbf {S} _{23}^{\mathrm {T} }\mathbf {S} _{23}\right).\end{aligned}}}$

These formulas may be used to determine the Cholesky factor after the insertion of rows or columns in any position, if the row and column dimensions are appropriately set (including to zero). The inverse problem,

${\displaystyle {\begin{aligned}{\tilde {\mathbf {A} }}&={\begin{pmatrix}\mathbf {A} _{11}&\mathbf {A} _{12}&\mathbf {A} _{13}\\\mathbf {A} _{12}^{\mathrm {T} }&\mathbf {A} _{22}&\mathbf {A} _{23}\\\mathbf {A} _{13}^{\mathrm {T} }&\mathbf {A} _{23}^{\mathrm {T} }&\mathbf {A} _{33}\\\end{pmatrix}}\end{aligned}}}$

with known Cholesky decomposition

${\displaystyle {\begin{aligned}{\tilde {\mathbf {S} }}&={\begin{pmatrix}\mathbf {S} _{11}&\mathbf {S} _{12}&\mathbf {S} _{13}\\0&\mathbf {S} _{22}&\mathbf {S} _{23}\\0&0&\mathbf {S} _{33}\\\end{pmatrix}}\end{aligned}}}$

and the desire to determine the Cholesky factor

${\displaystyle {\begin{aligned}\mathbf {L} &={\begin{pmatrix}\mathbf {L} _{11}&\mathbf {L} _{13}\\0&\mathbf {L} _{33}\\\end{pmatrix}}\end{aligned}}}$

of the matrix ${\displaystyle \mathbf {A} }$ with rows and columns removed,

${\displaystyle {\begin{aligned}\mathbf {A} &={\begin{pmatrix}\mathbf {A} _{11}&\mathbf {A} _{13}\\\mathbf {A} _{13}^{\mathrm {T} }&\mathbf {A} _{33}\\\end{pmatrix}},\end{aligned}}}$

yields the following rules:

${\displaystyle {\begin{aligned}\mathbf {L} _{11}&=\mathbf {S} _{11},\\\mathbf {L} _{13}&=\mathbf {S} _{13},\\\mathbf {L} _{33}&=\mathrm {chol} \left(\mathbf {S} _{33}^{\mathrm {T} }\mathbf {S} _{33}+\mathbf {S} _{23}^{\mathrm {T} }\mathbf {S} _{23}\right).\end{aligned}}}$

Notice that the equations above that involve finding the Cholesky decomposition of a new matrix are all of the form ${\displaystyle {\tilde {\mathbf {A} }}=\mathbf {A} \pm \mathbf {x} \mathbf {x} ^{*}}$, which allows them to be efficiently calculated using the update and downdate procedures detailed in the previous section.[22]

The above algorithms show that every positive definite matrix ${\displaystyle \mathbf {A} }$ 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 ${\displaystyle \mathbf {A} }$ is an ${\displaystyle n\times n}$ positive semi-definite matrix, then the sequence ${\textstyle \left(\mathbf {A} _{k}\right)_{k}:=\left(\mathbf {A} +{\frac {1}{k}}\mathbf {I} _{n}\right)_{k}}$ consists of positive definite matrices. (This is an immediate consequence of, for example, the spectral mapping theorem for the polynomial functional calculus.) Also,

${\displaystyle \mathbf {A} _{k}\rightarrow \mathbf {A} \quad {\text{for}}\quad k\rightarrow \infty }$

in operator norm. From the positive definite case, each ${\displaystyle \mathbf {A} _{k}}$ has Cholesky decomposition ${\displaystyle \mathbf {A} _{k}=\mathbf {L} _{k}\mathbf {L} _{k}^{*}}$. By property of the operator norm,

${\displaystyle \|\mathbf {L} _{k}\|^{2}\leq \|\mathbf {L} _{k}\mathbf {L} _{k}^{*}\|=\|\mathbf {A} _{k}\|\,.}$

The ${\displaystyle \leq }$ holds because ${\displaystyle M_{n}(\mathbb {C} )}$ equipped with the operator norm is a C* algebra. So ${\displaystyle \left(\mathbf {L} _{k}\right)_{k}}$ is a bounded set in the Banach space of operators, therefore relatively compact (because the underlying vector space is finite-dimensional). Consequently, it has a convergent subsequence, also denoted by ${\displaystyle \left(\mathbf {L} _{k}\right)_{k}}$, with limit ${\displaystyle \mathbf {L} }$. It can be easily checked that this ${\displaystyle \mathbf {L} }$ has the desired properties, i.e. ${\displaystyle \mathbf {A} =\mathbf {L} \mathbf {L} ^{*}}$, and ${\displaystyle \mathbf {L} }$ is lower triangular with non-negative diagonal entries: for all ${\displaystyle x}$ and ${\displaystyle y}$,

${\displaystyle \langle \mathbf {A} x,y\rangle =\left\langle \lim \mathbf {A} _{k}x,y\right\rangle =\langle \lim \mathbf {L} _{k}\mathbf {L} _{k}^{*}x,y\rangle =\langle \mathbf {L} \mathbf {L} ^{*}x,y\rangle \,.}$

Therefore, ${\displaystyle \mathbf {A} =\mathbf {L} \mathbf {L} ^{*}}$. Because the underlying vector space is finite-dimensional, all topologies on the space of operators are equivalent. So ${\displaystyle \left(\mathbf {L} _{k}\right)_{k}}$ tends to ${\displaystyle \mathbf {L} }$ in norm means ${\displaystyle \left(\mathbf {L} _{k}\right)_{k}}$ tends to ${\displaystyle \mathbf {L} }$ entrywise. This in turn implies that, since each ${\displaystyle \mathbf {L} _{k}}$ is lower triangular with non-negative diagonal entries, ${\displaystyle \mathbf {L} }$ is also.

Let ${\displaystyle \mathbf {A} }$ be a positive semi-definite Hermitian matrix. Then it can be written as a product of its square root matrix, ${\displaystyle \mathbf {A} =\mathbf {B} \mathbf {B} ^{*}}$. Now QR decomposition can be applied to ${\displaystyle \mathbf {B} ^{*}}$, resulting in ${\displaystyle \mathbf {B} ^{*}=\mathbf {Q} \mathbf {R} }$ , where ${\displaystyle \mathbf {Q} }$ is unitary and ${\displaystyle \mathbf {R} }$ is upper triangular. Inserting the decomposition into the original equality yields ${\displaystyle A=\mathbf {B} \mathbf {B} ^{*}=(\mathbf {QR} )^{*}\mathbf {QR} =\mathbf {R} ^{*}\mathbf {Q} ^{*}\mathbf {QR} =\mathbf {R} ^{*}\mathbf {R} }$. Setting ${\displaystyle \mathbf {L} =\mathbf {R} ^{*}}$ completes the proof.

• Sur la résolution numérique des systèmes d'équations linéaires, Cholesky's 1910 manuscript, online and analyzed on BibNum (in French and English) [for English, click 'A télécharger']

• "Cholesky factorization", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Cholesky Decomposition, The Data Analysis BriefBook
• Cholesky Decomposition on www.math-linux.com
• Cholesky Decomposition Made Simple on Science Meanderthal

• LAPACK is a collection of FORTRAN subroutines for solving dense linear algebra problems (DPOTRF, DPOTRF2, details performance)
• ALGLIB includes a partial port of the LAPACK to C++, C#, Delphi, Visual Basic, etc. (spdmatrixcholesky, hpdmatrixcholesky)
• libflame is a C library with LAPACK functionality.
• Notes and video on high-performance implementation of Cholesky factorization at The University of Texas at Austin.
• Cholesky : TBB + Threads + SSE is a book explaining the implementation of the CF with TBB, threads and SSE (in Spanish).
• library "Ceres Solver" by Google.
• LDL decomposition routines in Matlab.
• Armadillo is a C++ linear algebra package
• Rosetta Code is a programming chrestomathy site. on page topic.
• AlgoWiki is an open encyclopedia of algorithms’ properties and features of their implementations on page topic
• Intel® oneAPI Math Kernel Library Intel-Optimized Math Library for Numerical Computing ?potrf, ?potrs

• Generating Correlated Random Variables and Stochastic Processes, Martin Haugh, Columbia University

• Online Matrix Calculator Performs Cholesky decomposition of matrices online.