mathematics//linear algebra//determinant

The determinant is a single number computed from a square matrix that gives the factor by which its transformation multiplies areas or volumes, with a sign that says whether orientation is kept or flipped, and in engineering it is mostly used as a quick classifier: does the matrix crush a dimension, and, for a 2 × 2 system, is the equilibrium a saddle. A matrix that maps the unit square onto a parallelogram of area 3 has determinant 3 (or −3 if the parallelogram comes out mirrored). If the determinant is zero, at least one dimension has been flattened, the rank is less than full and what was flattened cannot be recovered.

The determinant is a single number computed from a square matrix that gives the factor by which its transformation multiplies areas or volumes, with a sign that says whether orientation is kept or flipped, and in engineering it is mostly used as a quick classifier: does the matrix crush a dimension, and, for a 2 × 2 system, is the equilibrium a saddle. A matrix that maps the unit square onto a parallelogram of area 3 has determinant 3 (or −3 if the parallelogram comes out mirrored). If the determinant is zero, at least one dimension has been flattened, the rank is less than full and what was flattened cannot be recovered.

Its link to the eigenvalues is what makes it useful in dynamics: the determinant is their product, and the trace (the sum of the diagonal) is their sum. For a two-state system x˙=Ax\dot x=Axx˙=Ax those two numbers fix both eigenvalues, because they are the roots of

λ2−tr⁡(A) λ+det⁡(A)=0.\lambda^2-\operatorname{tr}(A)\,\lambda+\det(A)=0 .λ2−tr(A)λ+det(A)=0.

So the stability of a 2 × 2 linear system can be read without computing anything: stable when the trace is negative and the determinant positive. A negative determinant means the two eigenvalues are real with opposite signs, one direction attracting and the other repelling: a saddle, whatever the trace. A drone pitch loop or a pair of coupled tanks linearized around an operating point is classified in seconds this way, and the trace-determinant plane maps every case.

The sign has a physical reading. A rotation has determinant +1, a reflection −1; an orthogonal matrix that is supposed to be a body-to-Earth rotation and shows −1 has had an axis swapped somewhere in the frame conventions, a classic bug when mounting an IMU upside down.

The determinant is a poor test of near-singularity, which is its most common misuse. Scaling a 10 × 10 identity by 0.1 gives a determinant of 10−1010^{-10}10−10 for a matrix that is perfectly well behaved, while a 2 × 2 matrix with determinant 1 can be hopeless to invert. How close a matrix is to losing rank is measured by its condition number, never by the size of its determinant.

It is rarely computed by its textbook formula, whose cost explodes with size; libraries take it as the product of the pivots of an LU decomposition, and work with its logarithm (slogdet) when it would overflow, as in the log-likelihood of a Gaussian with a large covariance matrix.