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Matrix Mechanics Lab

Matrices are machines for moving vectors. This lab turns determinant, rank, row reduction, eigenvectors, and SVD into visible operations.

Matrix mechanics lab

Make linear algebra move

This is the prerequisite layer for PCA, covariance, random matrices, and high-dimensional probability.

A matrix is a function from vectors to vectors. The determinant measures area scale, rank measures dimension kept, and trace tracks total diagonal action.

determinant
1.48
trace
2.30
rank
2
transpose Aᵀ
[1.25-0.250.651.05]
Matrix as a linear mapA matrix moves the basis arrows. Every other point follows by the same recipe.input squaretransformed shapee1e2A e1A e2A=1.250.65-0.251.05column 1 = A e1 = [1.25, -0.25]column 2 = A e2 = [0.65, 1.05]
Controls under the diagram

Change the object while the board stays close.

Geometry
Matrix preset
Input shape
Why this matters next

A covariance matrix is also a linear map

Determinant, rank, trace, and transpose are not separate trivia. They describe how a matrix moves area, collapses dimension, accumulates diagonal action, and flips rows with columns.

TheoryVectors, Matrices, Linear MapsThe concept page for matrix multiplication as geometryTheoryEigenvalues and EigenvectorsStable directions, characteristic equations, and diagonalizationNext DemoHigh-Dimensional Probability LabUse SVD and spectra to understand random matrices

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