# 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:

```python
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.

## Related

- [Basic Calculus](basic-calculus.md)
- [Epipolar geometry](epipolar-geometry.md)
- [Triangulation](triangulation.md)
- [Math libraries](../level-02-getting-familiar/math-libraries.md)
- [Schur complement / Sparsity](../level-02-getting-familiar/schur-complement-sparsity.md)
- [3D-3D correspondence](../level-02-getting-familiar/3d-3d-correspondence.md)