Author Archives: Nick Higham

What Is the CS Decomposition?

The CS (cosine-sine) decomposition reveals close relationships between the singular value decompositions (SVDs) of the blocks an orthogonal matrix expressed in block form. In full generality, it applies when the diagonal blocks are not necessarily square. We focus here mainly … Continue reading

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What’s New in MATLAB R2020a and R2020b?

In this post I discuss new features in MATLAB R2020a and R2020b. As usual in this series, I focus on a few of the features most relevant to my work. See the release notes for a detailed list of the … Continue reading

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What Is the Singular Value Decomposition?

A singular value decomposition (SVD) of a matrix is a factorization where and are orthogonal, , where , and . Partition and . The are called the singular values of and the and are the left and right singular vectors. … Continue reading

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What Is the Complex Step Approximation?

In many situations we need to evaluate the derivative of a function but we do not have an explicit formula for the derivative. The complex step approximation approximates the derivative (and the function value itself) from a single function evaluation. … Continue reading

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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

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What is the Kronecker Product?

The Kronecker product of two matrices and (also called the tensor product) is the matrix1 In other words, is the block matrix with block . For example, Notice that the entries of comprise every possible product , which is not … Continue reading

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