Matrix Inversion Using Cholesky Decomposition

Abstract—In this paper we present a method for matrix inversion based on Cholesky decomposition with reduced number of operations by avoiding computation of intermediate results; further, we use fixed point simulations to compare the numerical accuracy of the method. Keywords-matrix, inversion, Cholesky, LDL. I. INTRODUCTION Matrix inversion techniques based on Cholesky decomposition and the related LDL decomposition are efficient techniques widely used for inversion of positive- definite/symmetric matrices across multiple fields. Existing matrix inversion algorithms based on Cholesky decomposition use either equation solving [3] or triangular matrix operations [4] with most efficient implementation requiring

operations. 

In this paper we propose an inversion algorithm which reduces the number of operations by 16-17% compared to the existing algorithms by avoiding computation of some known intermediate results. In section 2 of this paper we review the Cholesky and LDL decomposition techniques, and discuss solutions to linear systems based on them. In section 3 we review the existing matrix inversion techniques, and how they may be extended to non-Hermitian matrices. In section 4 we discuss the proposed matrix inversion method. II. CHOLESKY DECOMPOSITION If is a positive-definite Hermitian matrix, Cholesky decomposition factorises it into a lower triangular matrix and its conjugate transpose [3], [5] & [6].

… (1) Or equivalently, using an upper triangular matrix as

… (2) In a software implementation the upper triangular matrix is preferred as operations are row-wise and compatible with C programming language. The elements of , are given as follows. Diagonal elements:

 √    ∑   

… (3)

Upper triangular elements, i.e. :

( ∑

) … (4) Note that since older values of aii aren’t required for computing newer elements, they may be overwritten by the value of rii, hence, the algorithm may be performed in-place using the same memory for matrices A and R. Cholesky decomposition is of order and requires

operations. Matrix inversion based on Cholesky decomposition is numerically stable for well conditioned matrices. If , with is the linear system with
variables, and satisfies the requirement for Cholesky decomposition, we can rewrite the linear system as

… (5) By letting , we have

… (6) and

… (7) These equations are solved using backward-substitution and require

operations each for the solution. 

A. LDL Decomposition If is a symmetric matrix, LDL decomposition factorises it into a lower triangular matrix, a diagonal matrix and conjugate transpose of the lower triangular matrix [5].

… (8) Or equivalently, using an upper triangular matrix as

… (9) This decomposition eliminates the need for square-root operation. The elements of , and diagonal elements of the matrix are given as follows. Diagonal elements:

    ∑   

… (10) Upper triangular elements, i.e. :

( ∑

) … (11) When efficiently implemented, the complexity of the LDL decomposition is same as Cholesky decomposition. If , with is the linear system with
variables, and satisfies the requirement for LDL decomposition, we can rewrite the linear system as

… (12) By letting , we have

… (13) and

… (14) These equations are solved using backward-substitution and when efficiently implemented, require

operations each for 

the solution. III. EXISTING TECHNIQUES A. Equation Solving If , we may find , and by solving

  

… (15) Where is the ith column of the identity matrix of order
[3]. This equation may be solved using either Cholesky or LDL based method as described above depending on the properties of . In either case since is Hermitian, it is sufficient to solve for upper (or lower) half of and update the other half with the complex conjugate values as
for . Solving for the upper half of the matrix requires two triangular matrix solutions with

multiply operations each. 

The total number of multiply operations including the decomposition is

. B. Triangular Matrix Operations If , we may find the inverse of , using Cholesky decomposition, we have

… (16) This implies:

… (17)

 , and                   is computed as 

follows.

… (18) Where is the ith column of the identity matrix of o

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