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Blog postAn matrix is normal if , that is, if commutes with its conjugate transpose. Although the definition is simple to state, its significance is not immediately obvious.
The definition says that the inner product of the th and th columns equals the inner product of the th and th rows for all and . For this means that the th row and the th column have the same norm for all . This fact can easily be used to show that a normal triangular matrix must be diagonal. It then follows from the Schu12 hours ago Read more 
Blog postA logarithm of a square matrix is a matrix such that , where is the matrix exponential. Just as in the scalar case, the matrix logarithm is not unique, since if then for any integer . However, for matrices the nonuniqueness is complicated by the presence of repeated eigenvalues. For example, the matrix
is an upper triangular logarithm of the identity matrix for any , whereas the obvious logarithms are the diagonal matrices , for integers , , and . Notice that the repeated eigenvalue1 week ago Read more 
Blog postA QR factorization of a rectangular matrix with is a factorization with orthonormal and upper trapezoidal. The factor has the form , where is and upper triangular. Partitioning conformably with we have
There are therefore two forms of QR factorization:
is the full QR factorization, is the reduced (also called economysized, or thin) QR factorization. To prove the existence of a QR factorization note that if has full rank then is symmetric positive definite. Since implies ,2 weeks ago Read more 
Blog postThe Cayley–Hamilton Theorem says that a square matrix satisfies its characteristic equation, that is where is the characteristic polynomial. This statement is not simply the substitution “”, which is not valid since must remain a scalar inside the term. Rather, for an , the characteristic polynomial has the form
and the Cayley–Hamilton theorem says that
Various proofs of the theorem are available, of which we give two. The first is the most natural for anyone familiar with t3 weeks ago Read more 
Blog postThe CS (cosinesine) 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 on the most practically important case of square diagonal blocks.
Let be orthogonal and suppose that is even and is partitioned into four equally sized blocks:
Then there exist orthogonal matrices such t4 weeks ago Read more 
Blog postIn 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 many changes in MATLAB and its toolboxes.
Exportgraphics (R2020a) The exportgraphics function is very useful for saving to a file a tightly cropped version of a figure with the border white instead of gray. Simple usages are
exportgraphics(gca,'image.pdf') exportgraphics(gca,'image.jpg1 month ago Read more 
Blog postA 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. We have , . The matrix is unique but and are not. The form of is
Here is an example, in which the entries of have been specially chosen to give simple forms for the elements of the factors:
The power of the SVD is that it reveals a great deal of1 month ago Read more 
Blog postIn 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. The catch is that it involves complex arithmetic.
For an analytic function we have the Taylor expansion
where is the imaginary unit. Assume that maps the real line to the real line and that and are real. Then equating real and2 months ago Read more 
Blog postWhen 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 the inverse of the perturbed matrix .
Sherman–Morrison Formula We will begin with the simpler case of a rank perturbation: , where and are vectors, and we consider first the case where . We might expect that for some (co2 months ago Read more 
Blog postA 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, the block matrix viewpoint enables elegant proofs of results and facilitates the development and understanding of numerical algorithms.
A block matrix is defined in terms of a partitioning, which breaks a matrix into contiguous pieces. The most common2 months ago Read more
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The musthave compendium on applied mathematics
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