The inverse iteration algorithm requires solving a linear system or calculation of the inverse matrix. For non-structured matrices (not sparse, not Toeplitz,...) this requires O(n^{3}) operations.
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In linear algebra, the Moore–Penrose inverse is a matrix that satisfies some but not necessarily all of the properties of an inverse matrix. This article collects together a variety of proofs involving the Moore-Penrose inverse.
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A square matrix A is called invertible or non-singular if there exists a matrix B such that :AB=BA=I_n. If B exists, it is unique and is called the inverse matrix of A, denoted A^{-1}.
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A square matrix A is called invertible or non-singular if there exists a matrix B such that :AB = BA = I , where I is the n×n identity matrix with 1s on the main diagonal and 0s elsewhere. If B exists, it is unique and is called the inverse matrix of A, denoted A.
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It follows that the matrices over a ring form a ring, which is noncommutative except if and the ground ring is commutative. A square matrix may have a multiplicative inverse, called an inverse matrix. In the common case where the entries belong to a commutative ring , a matrix has an inverse if and only if its determinant has a multiplicative inverse in . The determinant of a product of square matrices is the product of the determinants of the factors.
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In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix is the most widely known generalization of the inverse matrix. It was independently described by E. H. Moore in 1920, Arne Bjerhammar in 1951, and Roger Penrose in 1955. Earlier, Erik Ivar Fredholm had introduced the concept of a pseudoinverse of integral operators in 1903. When referring to a matrix, the term pseudoinverse, without further specification, is often used to indicate the Moore–Penrose inverse.
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Arthur Cayley introduced matrix multiplication and the inverse matrix in 1856, making possible the general linear group. The mechanism of group representation became available for describing complex and hypercomplex numbers. Crucially, Cayley used a single letter to denote a matrix, thus treating a matrix as an aggregate object. He also realized the connection between matrices and determinants, and wrote "There would be many things to say about this theory of matrices which should, it seems to me, precede the theory of determinants".
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The notion of unimodular matrix of integers must be extended by calling unimodular a matrix over an integral domain whose determinant is a unit. This means that the determinant is invertible and implies that unimodular matrices are the invertible matrices such all entries of the inverse matrix belong to the domain. To have an algorithmic solution of linear systems, a solution for a single linear equation in two unknowns is clearly required. In the case of the integers, such a solution is provided by extended Euclidean algorithm.
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156 In the general case, where A^{-1} is a n-by-n matrix and u and v are arbitrary vectors of dimension n, the whole matrix is updated and the computation takes 3n^2 scalar multiplications.Update of the inverse matrix by the Sherman–Morrison formula If u is a unit column, the computation takes only 2n^2 scalar multiplications. The same goes if v is a unit column. If both u and v are unit columns, the computation takes only n^2 scalar multiplications.
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Before diving into the full construction of this spin on the Redheffer matrix variants R_n defined above, observe that this type of expansion is in many ways essentially just another variation of the usage of a Toeplitz matrix to represent truncated power series expressions where the matrix entries are coefficients of the formal variable in the series. Let's explore an application of this particular view of a (0,1) matrix as masking inclusion of summation indices in a finite sum over some fixed function. See the citations to the references and for existing generalizations of the Redheffer matrices in the context of general arithmetic function cases. The inverse matrix terms are referred to a generalized Mobius function within the context of sums of this type in.
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The great advantage of this abstract construction is that, besides being much more relevant to the dynamic analysis, it can be easily converted by algebraic operations into an Input-Output scheme. The coefficients of production of a vertically integrated model are basically a linear combination of the coefficients of production of an Input-Output model. This means that it is possible to obtain empirical values of the vertically integrated coefficients for an economy. We need only to obtain the values of the production coefficients on each industry (as it is commonly done by the different national account agencies) as well as the capital data at current prices; then we take the transposed inverse matrix of the estimated coefficients and we multiplied it by the vector of capital stocks.
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