Linear algebra refresher

A.1 Vectors and inner products

A vector is an ordered tuple of real numbers. The notation $x \in \mathbb{R}^{n}$ says $x$ has $n$ components, indexed $x_1, x_2, \ldots, x_n$. In robotics, the most frequent vectors you will write down are joint configurations $q \in \mathbb{R}^{7}$ for a Franka-class arm, end-effector poses $p \in \mathbb{R}^{6}$ when orientation is expressed as an axis-angle vector, and the seven-vector of end-effector deltas plus gripper command that OpenVLA (arXiv:2406.09246) emits at 5 Hz.

Two vectors $x, y \in \mathbb{R}^{n}$ have an inner product $\langle x, y \rangle = x^\top y = \sum_{i} x_i y_i$. The inner product is a scalar; geometrically it equals $\lVert x \rVert\, \lVert y \rVert \cos\theta$, where $\theta$ is the angle between $x$ and $y$ and $\lVert x \rVert = \sqrt{x^\top x}$ is the Euclidean norm. Three properties of inner products do load-bearing work in the body of the book. Bilinearity — linear in each argument when the other is fixed — is what makes a transformer’s attention pattern $\mathrm{softmax}(QK^\top / \sqrt{d})$ computable in matrix form. Symmetry, $\langle x, y \rangle = \langle y, x \rangle$, is why a Gram matrix $X^\top X$ is symmetric. Positive-definiteness, $\langle x, x \rangle \geq 0$ with equality only at $x = 0$, is what lets the loss function in §3.3 be bounded below by zero.

Three derived quantities you will see often. The Euclidean distance $\lVert x - y \rVert$ measures how far apart two points are in configuration space. The cosine similarity $\langle x, y \rangle / (\lVert x \rVert\, \lVert y \rVert)$ is the inner product after length normalization; CLIP-style contrastive losses (Radford et al., 2021, ICML 2021) compute cosine similarity between image and text embeddings and that is the entire learning signal. The projection of $x$ onto a unit vector $u$ is $(u^\top x)\, u$; projection onto the column space of a matrix is the operation a least-squares solver performs under the hood.

A.2 Matrices as linear maps

A matrix $A \in \mathbb{R}^{m \times n}$ is two objects at once. As data, it is an $m$-by-$n$ array of numbers, indexed $A_{ij}$ for $i \in \{1, \ldots, m\}$ and $j \in \{1, \ldots, n\}$. As a function, it is a linear map from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$: every vector $x \in \mathbb{R}^{n}$ has an image $Ax \in \mathbb{R}^{m}$, and the map satisfies $A(\alpha x + \beta y) = \alpha\, Ax + \beta\, Ay$ for any scalars. The second view is the one that matters; once you read $A$ as a function, every operation the rest of the book performs on it becomes intelligible.

Three matrix examples that recur. The manipulator Jacobian $J(q) \in \mathbb{R}^{6 \times 7}$ of §3.1 maps joint velocities $\dot q \in \mathbb{R}^{7}$ to end-effector velocities $\dot p = J(q)\, \dot q \in \mathbb{R}^{6}$. The covariance matrix $\Sigma \in \mathbb{R}^{n \times n}$ of a Gaussian random vector encodes the pairwise covariances of its components; it is symmetric and positive semi-definite, and the diagonal entries are the per-component variances. The weight matrix $W \in \mathbb{R}^{d_{\text{out}} \times d_{\text{in}}}$ of a fully connected layer maps a $d_{\text{in}}$-dimensional input embedding to a $d_{\text{out}}$-dimensional output embedding; the entire weight budget of OpenVLA — about seven billion numbers — is a stack of such matrices.

Matrix-matrix multiplication $C = AB$, where $A \in \mathbb{R}^{m \times k}$ and $B \in \mathbb{R}^{k \times n}$, produces $C \in \mathbb{R}^{m \times n}$ with entries $C_{ij} = \sum_{\ell} A_{i\ell}\, B_{\ell j}$. As linear maps, $AB$ is the composition of $B$ followed by $A$: $(AB)x = A(Bx)$. Composition is associative but not commutative, which is why the order of matrix factors in a transformer attention computation matters and why most GPU training time on a VLA is spent inside one of a small number of large matrix multiplies.

The transpose $A^\top \in \mathbb{R}^{n \times m}$ is the matrix with swapped indices, $(A^\top)_{ij} = A_{ji}$. The transpose appears ubiquitously because $\langle Ax, y \rangle = \langle x, A^\top y \rangle$ — moving a linear map between the two arguments of an inner product just transposes it — and that identity is the backbone of backpropagation, where the gradient of a loss with respect to an input of a layer is computed by multiplying the upstream gradient by the layer’s transposed weight.

A.3 Rank, null space, and the four fundamental subspaces

The rank of a matrix $A \in \mathbb{R}^{m \times n}$ is the dimension of its column space — the subspace of $\mathbb{R}^{m}$ spanned by its columns. Rank is at most $\min(m, n)$. When rank equals $\min(m, n)$ the matrix is full rank; otherwise it is rank-deficient and behaves degenerately under inversion.

The null space of $A$, written $\ker(A)$ or $\mathrm{null}(A)$, is $\{x \in \mathbb{R}^{n} : Ax = 0\}$. Its dimension is $n - \mathrm{rank}(A)$. For the manipulator Jacobian of a redundant arm (7-DOF arm, 6-DOF task space, so $J \in \mathbb{R}^{6 \times 7}$), the null space is a one-dimensional subspace of joint-velocity vectors that produce zero end-effector velocity — that is, the directions the joints can move without the gripper moving. Null-space control exploits this freedom to keep the elbow up while the gripper does work, and is one of the reasons 7-DOF arms are useful when 6-DOF ones would also reach the target.

The four fundamental subspaces — column space, null space, row space, left null space — are connected by the fundamental theorem of linear algebra: column space and left null space are orthogonal complements in $\mathbb{R}^{m}$; row space and null space are orthogonal complements in $\mathbb{R}^{n}$. The practical implication for the book is that a least-squares solution always exists and is unique up to a shift in the null space, and that the Moore-Penrose pseudoinverse — used by every numerical inverse-kinematics solver in §4.2 — picks the shift that minimizes the norm of $x$.

A.4 Eigenvalues and eigenvectors

An eigenvector of a square matrix $A \in \mathbb{R}^{n \times n}$ is a nonzero vector $v$ such that $Av = \lambda v$ for some scalar $\lambda$, the corresponding eigenvalue. Geometrically, $v$ is a direction along which $A$ acts as pure scaling. A symmetric matrix has $n$ orthogonal real eigenvectors and $n$ real eigenvalues; the spectral theorem writes $A = V \Lambda V^\top$ where $V$ stacks the eigenvectors in columns and $\Lambda$ is the diagonal matrix of eigenvalues.

Eigenvalues do interpretable work in three places in the book. In control, the eigenvalues of the closed-loop dynamics matrix tell you whether a controller is stable (all eigenvalues in the open left half of the complex plane, for continuous-time systems). In probability, the eigenvalues of a covariance matrix are the variances along the principal axes of the Gaussian; this is the math under principal component analysis. In optimization, the eigenvalues of the Hessian of a loss function are the local curvatures; a learning-rate schedule that works well on a flat eigenvalue spectrum will diverge on a steep one, which is one motivation for adaptive optimizers like Adam.

A diagonalizable matrix $A = V \Lambda V^{-1}$ is essentially trivial to power: $A^{k} = V \Lambda^{k} V^{-1}$, where $\Lambda^{k}$ is the diagonal matrix with $\lambda_i^{k}$ on its diagonal. This is the trick value iteration in §5.2 implicitly uses: the Bellman operator is contractive with rate $\gamma$, which is its spectral radius, and the iteration converges geometrically at exactly that rate.

A.5 Singular value decomposition

The singular value decomposition (SVD) is the workhorse of numerical linear algebra. Every matrix $A \in \mathbb{R}^{m \times n}$ — not just symmetric, not just square — admits a factorization $A = U \Sigma V^\top$ where $U \in \mathbb{R}^{m \times m}$ and $V \in \mathbb{R}^{n \times n}$ are orthogonal (columns orthonormal) and $\Sigma \in \mathbb{R}^{m \times n}$ is rectangular-diagonal with non-negative entries $\sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_{\min(m, n)} \geq 0$. The $\sigma_i$ are the singular values of $A$.

Three uses of SVD recur. First, the pseudoinverse: $A^{+} = V \Sigma^{+} U^\top$, where $\Sigma^{+}$ inverts the nonzero singular values and zeros the rest. The damped least squares IK solver of §4.2 is structurally a pseudoinverse with the small singular values truncated, which is what prevents joint velocities from blowing up near a Jacobian singularity. Second, low-rank approximation: the best rank-$k$ approximation of $A$ in the Frobenius norm is obtained by zeroing all but the top $k$ singular values; this is the math under LoRA (Hu et al., 2022, ICLR 2022) and every adapter-based fine-tuning technique in Chapter 16. Third, conditioning: the ratio $\sigma_1 / \sigma_{\min(m, n)}$ is the condition number of $A$, and a large condition number means the matrix is close to singular and inverting it amplifies noise.

A.6 Quadratic forms and positive-definiteness

A quadratic form is a function $f(x) = x^\top A x$ for some symmetric matrix $A$. The form is positive definite if $x^\top A x > 0$ for all nonzero $x$, positive semi-definite if $x^\top A x \geq 0$, and negative definite if the inequality reverses. The Hessian of the loss function in §3.5 is symmetric and locally positive semi-definite near a minimum; the inertia matrix $M(q)$ in §4.3’s manipulator equation is positive definite for every physically realizable configuration, which is exactly why the computed-torque controller from that section is well-posed.

Positive-definite matrices have all-positive eigenvalues, are invertible, and admit a Cholesky factorization $A = LL^\top$ with $L$ lower triangular. Cholesky is what every Gaussian sampler in the book uses internally: to sample $x \sim \mathcal{N}(0, A)$, draw $z \sim \mathcal{N}(0, I)$ and return $Lz$.

A.7 Rotation matrices, $\mathrm{SO}(3)$, and the rest of the rigid-body story

Robotics adds a piece of structured linear algebra that pure ML does not: rotations. A rotation matrix in three dimensions is a matrix $R \in \mathbb{R}^{3 \times 3}$ with $R^\top R = I$ and $\det R = +1$. The set of all such matrices is the special orthogonal group $\mathrm{SO}(3)$; it is a three-dimensional manifold inside the nine-dimensional space of $3 \times 3$ matrices.

The body of the book treats $\mathrm{SO}(3)$ mostly through proxy representations — axis-angle vectors, quaternions, Euler angles — because that is what neural-network policies emit. Three things to remember about all of them. They are not unique: every rotation has two quaternion representations $\pm q$, and twelve Euler-angle conventions exist. Linear interpolation between rotation representations is not rotation interpolation: averaging two quaternions component-wise and renormalizing is not the same as the geodesic between them on $\mathrm{SO}(3)$, and the difference shows up as wobble in §10.2’s Diffusion Policy outputs if the model parameterizes orientation wrong. The Jacobian of a rotation representation with respect to its parameters is non-trivial; the right object for differentiating through rotations is the exponential map from the Lie algebra $\mathfrak{so}(3)$ (skew-symmetric matrices) to $\mathrm{SO}(3)$. Murray, Li, and Sastry (1994) is the canonical reference for the Lie-group story; the body of the book uses the exponential map without proof.

A.8 Numerical linear algebra in two paragraphs

Almost every matrix operation in the body of the book is computed numerically by a library — NumPy, PyTorch, JAX — that uses BLAS underneath. The relevant practical knowledge is small. Floating-point matrix multiplication is associative in mathematics but not in code: $(AB)C$ and $A(BC)$ can differ at the last few bits, and a long chain of matrix multiplications can accumulate small errors that compound. This is why training in mixed precision (fp16 or bf16) sometimes diverges where fp32 does not, and why VLA training recipes are picky about which layers are kept in fp32.

Matrix inversion is almost always the wrong operation to call. Instead of computing $A^{-1} b$, solve the linear system $Ax = b$ directly with an LU or QR factorization; the result is the same and the conditioning is better. Pseudoinverses for non-square matrices are computed via SVD, not by $A^\top (A A^\top)^{-1}$ or analogues; the latter formula is correct on paper and numerically catastrophic when $A$ is poorly conditioned, which is the most common case in inverse-kinematics work near a singularity.

A.9 A quick reference of identities

A handful of identities are used so often in the body of the book that they are worth memorizing.

The chain rule for matrix-valued functions: if $f : \mathbb{R}^{n} \to \mathbb{R}^{m}$ and $g : \mathbb{R}^{m} \to \mathbb{R}^{p}$, then the Jacobian of the composition $g \circ f$ at $x$ is $J_{g \circ f}(x) = J_g(f(x))\, J_f(x)$. This is the matrix form of the chain rule and the mathematical backbone of backpropagation.

The matrix-multiplication identity for transposes: $(AB)^\top = B^\top A^\top$. Used implicitly every time gradients flow backward through a linear layer.

The Woodbury identity for low-rank updates of an inverse: $(A + UV^\top)^{-1} = A^{-1} - A^{-1} U (I + V^\top A^{-1} U)^{-1} V^\top A^{-1}$. This is the identity under LoRA’s training-time inference path, and the identity Kalman filters use to update a covariance matrix from a low-dimensional measurement.

The trace cyclicity property: $\mathrm{tr}(ABC) = \mathrm{tr}(BCA) = \mathrm{tr}(CAB)$. This is used in derivations of every loss whose gradient involves a trace, and it is the identity behind the “reparameterization trick” in §3.4’s discussion of VAEs.

The Frobenius inner product: $\langle A, B \rangle_F = \mathrm{tr}(A^\top B) = \sum_{ij} A_{ij} B_{ij}$. This is the inner product that makes the space of matrices into a vector space; the Frobenius norm $\lVert A \rVert_F = \sqrt{\langle A, A \rangle_F}$ is what “distance between weight matrices” almost always means.

Five identities, four sections of background, one quick-reference list — the rest is in the body of the book. If you forget what $A^\top A$ is, look here; if you forget why the singular values matter to a controller, look at §4.2 and then come back.