Linear Algebra on SciDraft | A Hugo Theme

Inner-Product Axioms (axiom)

Mon, 21 Sep 2026 15:22:00 +0900

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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”.

The Woodbury Identity (identity)

Mon, 21 Sep 2026 14:42:00 +0900

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.

Matrix Products Do Not Commute (counterexample)

Mon, 21 Sep 2026 12:12:00 +0900

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.

Sylvester's Criterion for Positive Definiteness (criterion)

Mon, 21 Sep 2026 09:42:00 +0900

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.

The Gram Matrix Is Positive Semi-Definite (property)

Mon, 21 Sep 2026 08:52:00 +0900

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.

Power Iteration with a Convergence Check (pseudocode)

Mon, 21 Sep 2026 08:39:00 +0900

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.