Linear Algebra

Demo note. This note demonstrates what the SciDraft Hugo theme can do for academic and scientific publishing — for research institutions, researchers and students. Its content is demonstration material only: readers should ignore whether the demo content is correct. Attention is built on a quantity that looks like a similarity but is defined by three axioms: an inner product is symmetric, linear in each argument, and positive definite. Those three properties are what allow a score to be read as “how much one vector points along another”. ...

Demo note. This note demonstrates what the SciDraft Hugo theme can do for academic and scientific publishing — for research institutions, researchers and students. Its content is demonstration material only: readers should ignore whether the demo content is correct. The Woodbury identity rewrites the inverse of a matrix plus a low-rank correction in terms of the inverse of the original matrix and a much smaller inverse. It is the algebraic fact behind every method that updates a large system cheaply. ...

Demo note. This note demonstrates what the SciDraft Hugo theme can do for academic and scientific publishing — for research institutions, researchers and students. Its content is demonstration material only: readers should ignore whether the demo content is correct. A counterexample is the cheapest way to kill a plausible statement, and the one worth having ready concerns the order of matrix multiplication. The tempting assertion that \(AB = BA\) fails for almost every pair of matrices, and it fails on the smallest possible example. ...

Demo note. This note demonstrates what the SciDraft Hugo theme can do for academic and scientific publishing — for research institutions, researchers and students. Its content is demonstration material only: readers should ignore whether the demo content is correct. Positive definiteness is the condition that decides whether a quadratic form curves upwards, and Sylvester’s criterion turns it into a finite computation: look at the determinants of the leading blocks of the matrix, from the top left outwards. ...

Demo note. This note demonstrates what the SciDraft Hugo theme can do for academic and scientific publishing — for research institutions, researchers and students. Its content is demonstration material only: readers should ignore whether the demo content is correct. The matrix of pairwise inner products of a set of vectors has a property that constrains what similarity patterns are possible: it is symmetric and positive semi-definite, so its eigenvalues are non-negative and its diagonal entries are the squared lengths of the vectors. ...

Demo note. This note demonstrates what the SciDraft Hugo theme can do for academic and scientific publishing — for research institutions, researchers and students. Its content is demonstration material only: readers should ignore whether the demo content is correct. The largest singular value of a weight matrix is one of the numbers used to judge how a layer transforms its input, and it is usually estimated rather than computed: repeated multiplication by the matrix makes any starting vector align with the leading eigenvector, after which the Rayleigh quotient gives the value. ...