The QR Algorithm Computes Eigenvalues and Singular Values

Posted by Cleve Moler,

The QR algorithm is one of the world's most successful algorithms. We can use animated gifs to illustrate three variants of the algorithm, one for computing the eigenvalues of a nonsymmetric matrix, one for a symmetric matrix, and one for the singular values of a rectangular matrix. In all three cases, the QR iteration itself is preceded by a reduction to a compact form. All transformations are orthogonal similarities using Givens and Householder transformations. These are numerically stable, preserve eigenvalues, and preserve any symmetry.... read more >>

Hadamard Matrices

Posted by Cleve Moler,

I have just returned from the ICIAM2019 conference in Valencia, Spain. It was a huge conference -- 4,000+ attendees, dozens of prize and invited talks, hundreds of parallel minisympsia. I gave a talk in a two-part minisymposium organized by Nick Higham and Rob Corless. I outlined the first part of the talk in this blog a month ago. This post outlines the second part, which was about Hadamard matrices. Some of it is taken from a post in this blog five years ago.... read more >>

The Reuleaux Triangle and Curves of Constant Width

Posted by Cleve Moler,

Why are manhole covers round? It is so they won't fall through the hole they are intended to cover. They have the same diameter regardless of where it is measured. If the hole has a slightly smaller diameter, it is not possible to orient the cover so that it will fall through. A square or rectangular cover can be turned slightly and it will easily fit through the hole.... read more >>

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