Matrix Fundamentals

Review matrices as linear maps, focusing on rank, determinant, and factorization tools used across the control stack.

Matrices package a linear map T:Rn→RmT: \mathbb{R}^n \to \mathbb{R}^m into a rectangular array that transforms basis coordinates. A matrix A∈Rm×nA \in \mathbb{R}^{m \times n} acts on a vector x\mathbf{x} by stacking weighted combinations of components, Ax=∑j=1nxjajA\mathbf{x} = \sum_{j=1}^n x_j \mathbf{a}_j, where aj\mathbf{a}_j is the jj-th column. The column space captures all reachable combinations, and its dimension is the rank abla abla, which indicates how many independent directions are preserved. Rank deficiency points to constraints or redundant actuators, themes that show up when forming defects in Pontryagin's Minimum Principle Notes or assembling observers in Kalman Filter Essentials.

Key invariants such as the determinant measure volume scaling: det⁡(A)\det(A) tells us how hypercubes warp under AA. If det⁡(A)=0\det(A) = 0, the transformation collapses volume, signaling loss of invertibility. Decompositions like QRQR or singular value decomposition (SVD) reveal orthogonal bases and stretch factors that we later use to precondition Line Search Methods or to regularize Hessians in Nonlinear MPC Overview. The spectral theorem gives us diagonalization for symmetric matrices, unlocking tools like modal analysis in Kinematic Bicycle Model.

import numpy as np

A = np.array([[1.0, 2.0],
              [3.0, 4.0]])
values, vectors = np.linalg.eig(A)
print("Eigenvalues:", values)
print("Eigenvectors:\n", vectors)

Conditioning matters whenever we invert matrices or solve linear systems. The 2-norm condition number κ2(A)=∥A∥2∥A−1∥2\kappa_2(A) = \lVert A \rVert_2 \lVert A^{-1} \rVert_2 bounds how errors in the right-hand side translate into solution errors. When κ2(A)\kappa_2(A) is large, small measurement noise can corrupt state estimates, motivating the covariance inflation heuristics discussed in EKF and UKF Overview. For control design, recognizing structure (bandedness, sparsity, symmetry) enables tailored solvers exploited in Direct Transcription.

See also