Basic Linear Algebra

Linear algebra is the working language of SLAM: points, poses, residuals, and observations are all vectors and matrices, and every solver ultimately reduces to matrix computations.

Vectors and Matrices

A vector $\mathbf{v} \in \mathbb{R}^n$ is a column of $n$ real numbers. In SLAM, vectors represent points, translations, velocities, and residuals. A matrix $A \in \mathbb{R}^{m \times n}$ represents a linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$ . The central object in SLAM geometry is the matrix-vector product $A\mathbf{x} = \mathbf{b}$ .

Determinant

The determinant $\det(A)$ of a square matrix measures the signed volume scaling factor of the linear map $A$ . Key facts:

$\det(A) \neq 0 \Leftrightarrow A$ is invertible.
For a rotation matrix $R \in SO(3)$ : $\det(R) = +1$ .
$\det(AB) = \det(A)\det(B)$ .

Dot Product and Cross Product

The dot product of $\mathbf{u}, \mathbf{v} \in \mathbb{R}^3$ is $\mathbf{u} \cdot \mathbf{v} = \mathbf{u}^T\mathbf{v} = \|\mathbf{u}\|\|\mathbf{v}\|\cos\theta$ ; it is used for projections and checking orthogonality. The cross product $\mathbf{u} \times \mathbf{v}$ produces a vector perpendicular to both, with magnitude $\|\mathbf{u}\|\|\mathbf{v}\|\sin\theta$ . It appears in the skew-symmetric matrix formulation of the essential matrix and in computing surface normals.

The cross product can be written as a matrix-vector product using the skew-symmetric matrix:

$\mathbf{u} \times \mathbf{v} = [\mathbf{u}]_\times \mathbf{v}, \qquad [\mathbf{u}]_\times = \begin{bmatrix} 0 & -u_3 & u_2 \\ u_3 & 0 & -u_1 \\ -u_2 & u_1 & 0 \end{bmatrix}$

This little identity is everywhere in SLAM: the essential matrix is $E = [\mathbf{t}]_\times R$ , and Lie algebra elements of $\mathfrak{so}(3)$ are skew-symmetric matrices.

Rank, Inverse, and Transpose

The rank of $A$ is the dimension of its column space. In SLAM, the fundamental matrix has rank 2 by construction, and a point cloud matrix is rank-deficient when all points are coplanar (a degenerate configuration for SLAM initialization). The inverse $A^{-1}$ satisfies $AA^{-1} = I$ and exists iff $\det(A) \neq 0$ . For rotation matrices, $R^{-1} = R^T$ (orthogonality) — exploited constantly to avoid computing a full inverse.

A quick NumPy check that a matrix is a valid rotation:

import numpy as np

theta = np.pi / 4
R = np.array([[np.cos(theta), -np.sin(theta), 0],
              [np.sin(theta),  np.cos(theta), 0],
              [0,              0,             1]])

print(np.allclose(R.T @ R, np.eye(3)))  # True: orthogonal
print(np.linalg.det(R))                 # 1.0: proper rotation

Solving linear systems

SLAM back-ends never invert matrices explicitly; they factorize and solve:

Cholesky factorization ( $A = LL^T$ ) for symmetric positive-definite systems — the normal equations $J^TJ\,\Delta\mathbf{x} = -J^T\mathbf{e}$ of bundle adjustment are solved this way (sparse Cholesky in practice).
QR factorization is the numerically safer alternative for least squares, working on $J$ directly without forming $J^TJ$ (which squares the condition number).
SVD is the most robust (and most expensive) option, and the only one that cleanly handles rank-deficient, homogeneous systems $A\mathbf{x} = \mathbf{0}$ .

Singular Value Decomposition (SVD)

SVD is the most important matrix factorization in SLAM. Any matrix $A \in \mathbb{R}^{m \times n}$ decomposes as

$A = U \Sigma V^T$

where $U$ and $V$ are orthogonal and $\Sigma$ is diagonal with non-negative singular values $\sigma_1 \geq \sigma_2 \geq \cdots \geq 0$ . Geometrically, any linear map is a rotation/reflection, followed by a coordinate-wise scaling, followed by another rotation/reflection. Uses in SLAM:

Essential matrix decomposition: SVD of $E$ yields the four candidate relative poses $[R|\pm\mathbf{t}]$ .
DLT triangulation: the solution is the right singular vector of $A$ with the smallest singular value.
ICP alignment: SVD of the cross-covariance matrix $H = \sum_i \mathbf{p}_i \mathbf{q}_i^T$ gives the optimal rotation.

The ratio $\sigma_1/\sigma_n$ is the condition number: it measures how much a linear solve amplifies input noise. Ill-conditioned systems (huge condition numbers) are why coordinate normalization matters in the 8-point algorithm and DLT.

Eigenvalues and Eigenvectors

A non-zero vector $\mathbf{v}$ and scalar $\lambda$ satisfying $A\mathbf{v} = \lambda\mathbf{v}$ are an eigenvector and eigenvalue of $A$ . The Harris corner detector classifies image patches from the eigenvalues of the structure tensor, and PCA of a point cloud uses eigenvectors of the covariance matrix to find principal directions (e.g., fitting a plane). For symmetric matrices (covariances, structure tensors, $J^TJ$ ) the eigenvalues are real and the eigenvectors orthogonal — and eigendecomposition coincides with SVD.

Common pitfalls

Explicitly inverting matrices (inv(A) @ b) instead of solving (np.linalg.solve, Cholesky): slower and less accurate.
Forgetting normalization before solving homogeneous systems — DLT and the 8-point algorithm degrade badly on raw pixel coordinates.
Sign/reflection ambiguity in SVD-based rotation recovery: always check $\det(R) = +1$ and correct with a sign flip if you got a reflection.

Why it matters for SLAM

Every stage of a SLAM pipeline is linear algebra in disguise: projecting points uses matrix products, minimal solvers (8-point, DLT, ICP) reduce to SVD, and the normal equations of bundle adjustment are a large sparse linear system. Fluency here — especially with SVD, orthogonal matrices, and rank — is what lets you read SLAM papers and debug geometry code.