Completion method

2020 Mathematics Subject Classification: Primary: 65F05 [MSN][ZBL]

A method for calculating the inverse of a matrix, based on a recurrent transition which involves the calculation of a matrix $(C+uv{)}^{-1}$, where $u$ is a column vector, $v$ is a row vector, by the formula $$(C+uv{)}^{-1}=C^{-1}-\frac{1}{\gamma }{C}^{-1}uv{C}^{-1},\gamma =1+v{C}^{-1}u.$$

The computational scheme of the method is as follows. Let $A=(A_{ij})$ be a given matrix of order $n$. Consider a sequence $A_0=I,A_1,\dots ,A_n$, where $A_k=A_{k-1}+e_k{a}_k$ and $e_k$ is the $k$-th column of the identity matrix $I$, $a_k=(a_{k1},\dots ,a_{k,k-1},a_{kk}-1,a_{k,k+1},\dots ,a_{kn})$. Then $A_n=A$ and the matrix $A^{-1}$ is obtained by applying the above-described procedure $n$ times. The computational formulas in this case are the following: If $a_j^{(k)}$ is the $j$-th column of $A_k$, then for $k=1,\dots ,n$, $$a_j^{(k)}=a_j^{(k-1)}-\frac{a_k{a}_j^{(k-1)}}{1+a_k{a}_k^{(k-1)}}{a}_k^{(k-1)},j=1,\dots ,n.$$

It is sufficient to compute the elements of the first $k$ rows of the matrix $A_k^{-1}$, since all subsequent rows coincide with the rows of the identity matrix.

Other possibilities of arranging the computations in the completion method based on certain modifications of (*) are known, e.g. the so-called Ershov method (see [1]).

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This method is also called the bordering method (cf. [1]). See, however, also Bordering method.