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Monthly Archives: September 2020
What Is the Sherman–Morrison–Woodbury Formula?
When a nonsingular matrix is perturbed by a matrix of rank , the inverse also undergoes a rank perturbation. More precisely, if has rank and is nonsingular then the identity shows that The Sherman–Morrison–Woodbury formula provides an explicit formula for … Continue reading
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What Is a Block Matrix?
A matrix is a rectangular array of numbers treated as a single object. A block matrix is a matrix whose elements are themselves matrices, which are called submatrices. By allowing a matrix to be viewed at different levels of abstraction, … Continue reading
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What Is a Householder Matrix?
A Householder matrix is an orthogonal matrix of the form It is easily verified that is orthogonal (), symmetric (), involutory ( that is, is a square root of the identity matrix), where the last property follows from the first … Continue reading
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What is a Sparse Matrix?
A sparse matrix is one with a large number of zero entries. A more practical definition is that a matrix is sparse if the number or distribution of the zero entries makes it worthwhile to avoid storing or operating on … Continue reading
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What Is the Sylvester Equation?
The Sylvester equation is the linear matrix equation where , , and . It is named after James Joseph Sylvester (1814–1897), who considered the homogeneous version of the equation, in 1884. Special cases of the equation are (a standard linear … Continue reading