See how 2x2 matrices transform the plane. Drag basis vectors and watch the grid deform. Shows determinant, eigenvalues, and SVD.
Choose a 2x2 matrix or drag the transformed basis vectors to see the corresponding linear map. The visualizer moves the coordinate grid and reports the determinant, eigenvalues, and singular value decomposition associated with that map. It is useful for connecting matrix entries to geometry rather than reading the numbers in isolation. A common mistake is treating the matrix rows as basis-vector endpoints. Under the usual column-vector convention, its columns are where the standard basis vectors land.
A concrete input and expected output from the current implementation.
Input
Matrix [[2, 0], [0, 1]]
Expected output
e1 maps to (2, 0), e2 maps to (0, 1), determinant = 2, eigenvalues = 2 and 1, and singular values = 2 and 1. The grid is stretched horizontally by a factor of 2 and is unchanged vertically.
The transformation sends (x, y) to (2x, y). A unit square becomes a 2-by-1 rectangle, so its area is multiplied by 2; the two coordinate axes are also eigenvector directions with scale factors 2 and 1.
For A = [[a, b], [c, d]], each point (x, y) maps to (ax + by, cx + dy). The first column is the image of e1 = (1, 0), and the second is the image of e2 = (0, 1). Grid lines remain straight, parallel lines remain parallel, and the origin remains fixed, although the plane can collapse onto a line or point.
The determinant ad - bc is the signed area scale: its magnitude gives area change, its sign records orientation, and zero indicates collapse to lower dimension. Eigenvalues describe scale factors along corresponding eigenvector directions. The SVD writes the matrix as U Sigma V^T, separating orthogonal changes of direction from the nonnegative axis stretches listed in Sigma.
No, not necessarily. A nonzero 2x2 matrix with determinant zero can have rank one, sending the entire plane onto a single line through the origin. Only the zero matrix sends every vector to the origin. The deformed grid may therefore appear flattened into a line rather than reduced to one point.
Eigenvalues measure scaling only along eigenvector directions, which need not be perpendicular and may not exist as real directions. Singular values measure the largest and smallest orthogonal stretch factors of the whole transformation. For a shear, the eigenvalues can both equal 1 even though some vectors visibly lengthen and the singular values differ from 1.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
Calculate percentages, percentage change, and what percent one number is of another with step-by-step breakdowns.
Calculate tips and split bills between multiple people with customizable tip percentages and rounding options.
Calculate monthly payments, total interest, and generate amortization schedules for loans and mortgages.
Calculate Body Mass Index from height and weight with category classification and healthy range guidance.
Calculate sale prices, savings amounts, and final costs after single or stacked percentage discounts.
Compute mean, median, mode, standard deviation, variance, and other descriptive statistics from a data set.