秩序之下 · No.15 · 实时图版

Linear

线性代数

Plates on matrices and what they do, each one computed in the browser as you watch: linear maps, a sheet of graph paper moved by a two-by-two matrix, whose columns are where the two basis vectors land, whose determinant is the factor by which it multiplies every area, falling through zero as the sheet is pressed onto a line and turned over, and whose eigenvectors are the lines it leaves in place; the cat map, an integer matrix of determinant one applied to a picture on a torus, which stretches it into stripes and then into what looks like noise, until on a grid of 610 by 610 points it brings the picture back, turned half a turn after fifteen steps and exactly after thirty; least squares, the line through three points that makes the total area of the squares on its errors as small as it can be, which in the space of the data is the point of a plane nearest to a point off it, found by a sphere that shrinks until it just touches the plane, the error standing at right angles to both columns of the matrix, and, with the errors measured at right angles to the line, the principal axis of a cloud of points; principal axes, the perpendicular eigenvectors of a symmetric matrix, where its quadratic form is largest and smallest on the unit circle and along which the matrix turns diagonal, with the signs of its eigenvalues deciding whether xᵀSx = 1 is an ellipse or a hyperbola, an ellipsoid or a hyperboloid of one sheet or of two; and singular values, the semi-axes of the ellipse any matrix makes of a circle, the matrix as a turn, a stretch and a turn, the unit sphere pressed flat as the rank falls, and the cat of the second plate rebuilt term by term from the closest matrices of each rank. The corner writes the matrix of the moment, and reads each quantity twice, once from its entries and once from the picture.