Nicholas J. Higham
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About Nicholas J. Higham
Nick Higham is the Richardson Professor of Applied Mathematics at the University of Manchester. His research interests focus on numerical analysis, ranging from theory to the development of algorithms and software. He is a keen expositor and blogs at http://nickhigham.wordpress.com. Nick was elected a Fellow of the Royal Society in 2007.
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Blog postAn circulant matrix is defined by parameters, the elements in the first row, and each subsequent row is a cyclic shift forward of the one above:
Circulant matrices have the important property that they are diagonalized by the discrete Fourier transform matrix
which satisfies , so that is a unitary matrix. ( is a Vandermonde matrix with points the roots of unity.) Specifically,
Hence circulant matrices are normal (). Moreover, the eigenvalues are given by ,
N6 days ago Read more 
Blog postThe conference Advances in Numerical Linear Algebra: Celebrating the 60th Birthday of Nick Higham was held at the University of Manchester, July 6–8, 2022.
Most of the talks are available on the NLA Group YouTube channel and links to them are available on the conference web page.
Here is the conference photo.
(Hires version)
Here is me with some of my current and former PhD students.
(Hires version)
And with some of my current and former postdo1 week ago Read more 
Blog postThe definition of matrix multiplication says that for matrices and , the product is given by . Each element of is an inner product of a row of and a column of , so if this formula is used then the cost of forming is additions and multiplications, that is, operations. For over a century after the development of matrix algebra in the 1850s by Cayley, Sylvester and others, all methods for matrix multiplication were based on this formula and required operations.
In 1969 Volker Strassen sh2 weeks ago Read more 

Blog postSixteen members of the Numerical Linear Algebra group at the University of Manchester recently attended a twoday creativity workshop in order to generate ideas for our research and other activities. The workshop was facilitated by Dennis Sherwood, who is an expert in creativity and has run many such workshops. Dennis and I have previously collaborated on workshops for the Manchester Numerical Analysis group (2013), the EPSRC NAHPC Network (2014), and the SIAM leadership (2018).
A cr2 months ago Read more 
Blog postIn a displayed mathematical equation with more than one component, how much space should be placed between the components?
Here are the guidelines I use, with examples in LaTeX. Recall that a \quad is approximately the width of a capital M and \qquad is twice the width of a \quad.
Case 1. Equation with Qualifying Expression An equation or other mathematical construct is separated from a qualifying expression by a \quad. Examples:
When the qualifying expression is a p3 months ago Read more 
Blog postThe Pascal matrix is the symmetric matrix defined by
It contains the rows of Pascal’s triangle along the antidiagonals. For example:
In MATLAB, the matrix is pascal(n).
The Pascal matrix is positive definite and has the Cholesky factorization
where the rows of are the rows of Pascal’s triangle. For example,
From (1) we have . Form this equation, or by inverting (1), it follows that has integer elements. Indeed the inverse is known to have element <3 months ago Read more 
Blog postSix matrix factorizations dominate in numerical linear algebra and matrix analysis: for most purposes one of them is sufficient for the task at hand. We summarize them here.
For each factorization we give the cost in flops for the standard method of computation, stating only the highest order terms. We also state the main uses of each factorization.
For full generality we state factorizations for complex matrices. Everything translates to the real case with “Hermitian” and “un4 months ago Read more 
Blog postA Schur decomposition of a matrix is a factorization , where is unitary and is upper triangular. The diagonal entries of are the eigenvalues of , and they can be made to appear in any order by choosing appropriately. The columns of are called Schur vectors.
A subspace of is an invariant subspace of if for all . If we partition and conformably we can write
which gives , showing that the columns of span an invariant subspace of . Furthermore, . The first column of is an eigenve5 months ago Read more 
Blog postA permutation matrix is a square matrix in which every row and every column contains a single and all the other elements are zero. Such a matrix, say, is orthogonal, that is, , so it is nonsingular and has determinant . The total number of permutation matrices is .
Premultiplying a matrix by reorders the rows and postmultiplying by reorders the columns. A permutation matrix that has the desired reordering effect is constructed by doing the same operations on the identity matrix.
5 months ago Read more 
Blog postIn this post I discuss some of the new features in MATLAB R2022a, focusing on ones that relate to my particular interests. See the release notes for a detailed list of the many changes in MATLAB and its toolboxes. For my articles about new features in earlier releases, see here.
Themes MATLAB Online now has themes, including a dark theme (which is my preference). We will have to wait for a future release for themes to be supported on desktop MATLAB.
I recall that @nhigham was5 months ago Read more
Titles By Nicholas J. Higham
The musthave compendium on applied mathematics
This is the most authoritative and accessible singlevolume reference book on applied mathematics. Featuring numerous entries by leading experts and organized thematically, it introduces readers to applied mathematics and its uses; explains key concepts; describes important equations, laws, and functions; looks at exciting areas of research; covers modeling and simulation; explores areas of application; and more.
Modeled on the popular Princeton Companion to Mathematics, this volume is an indispensable resource for undergraduate and graduate students, researchers, and practitioners in other disciplines seeking a userfriendly reference book on applied mathematics.
 Features nearly 200 entries organized thematically and written by an international team of distinguished contributors
 Presents the major ideas and branches of applied mathematics in a clear and accessible way
 Explains important mathematical concepts, methods, equations, and applications
 Introduces the language of applied mathematics and the goals of applied mathematical research
 Gives a wide range of examples of mathematical modeling
 Covers continuum mechanics, dynamical systems, numerical analysis, discrete and combinatorial mathematics, mathematical physics, and much more
 Explores the connections between applied mathematics and other disciplines
 Includes suggestions for further reading, crossreferences, and a comprehensive index