mathematics//linear algebra//null space

The null space of a matrix is the set of input vectors it sends to zero, everything the transformation destroys, and it is where an engineer looks to find what a sensor suite cannot see, what a set of equations cannot decide and which motions an actuator layout leaves free. It is a subspace: if two vectors vanish under \(A\), so does any combination of them. Its dimension is what the rank leaves over, so a matrix with \(n\) columns and rank \(r\) has a null space of dimension \(n-r\).

The null space of a matrix is the set of input vectors it sends to zero, everything the transformation destroys, and it is where an engineer looks to find what a sensor suite cannot see, what a set of equations cannot decide and which motions an actuator layout leaves free. It is a subspace: if two vectors vanish under AAA, so does any combination of them. Its dimension is what the rank leaves over, so a matrix with nnn columns and rank rrr has a null space of dimension n−rn-rn−r.

N(A)={x:Ax=0},dim⁡N(A)=n−rank⁡(A)\mathcal N(A)=\{x : Ax = 0\},\qquad \dim\mathcal N(A)=n-\operatorname{rank}(A)N(A)={x:Ax=0},dimN(A)=n−rank(A)

A vector in the null space is invisible at the output, and adding it to any input changes nothing that comes out. That single fact explains two practical situations. In a linear system of equations with fewer equations than unknowns, every solution plus any null-space vector is still a solution, so the data cannot choose among them and a criterion (smallest norm, a prior) has to. In estimation, stack what the sensors see over the first nnn steps into the observability matrix OOO: a state xxx with Ox=0Ox=0Ox=0 produces exactly the same measurements as the zero state, forever.

The unobservable directions are the null space of the observability matrix.

They are states the measurement chain turns into nothing, and a filter left to estimate them lets them drift freely (observability).

The examples are concrete. A gyroscope integrated for attitude with no accelerometer to compare against leaves its own bias in the null space: a constant offset in the rate produces the same readings as a slow true rotation. Visual-inertial odometry has a four-dimensional null space (absolute position and rotation about gravity), which is why it reports where it has travelled and never where it started.

Exactly zero is the textbook case; nearly zero is the field case. Directions that a matrix shrinks by a tiny singular value are not in the null space, yet their noise is amplified enormously when the matrix is inverted. A short window of position readings sees velocity this way, through a difference divided by Δt\Delta tΔt.

A null space is not always bad news. In a control allocation with more actuators than axes to control, the null space holds the combinations of motor commands that produce no net force or torque, and the allocator spends that freedom to keep motors away from saturation.