Trace — where it appears
Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.
![The polynomial whose roots are the stretches. The determinant of A − λI plotted against λ for the map [2, 1, 1, 2], with its roots at 3 and 1 marked.](/f/the-polynomial-whose-roots-are-the-stretches-5281ff59.svg)
The polynomial whose roots are the stretches
Finding the directions a map leaves alone means finding the numbers at which it crushes something to nothing. Those numbers are the roots of one quadratic, and everything the map does is written in its two coefficients.
![The flow of a linear equation, and the matrix that runs it for one unit of time. Paths of points moving so that their velocity is [0.25, −1.2, 1.2, 0.25] applied to their position, with the position after time 1 marked on each; the matrix taking start to finish is e^A.](/f/the-flow-of-a-linear-equation-and-the-matrix-that-runs-it-15ef5202.svg)
The exponential of a square
The series for e makes perfect sense with a matrix in it. What comes out solves a system of equations the way the ordinary exponential solves one, and a skew matrix exponentiates into a rotation with no trigonometry anywhere.
When the label may be a matrix
A permutation carries exactly one bit into any commutative target, and commutativity is the restriction doing all the work. Drop it — let the label be a matrix — and what survives is a short finite table, computed here from traces and checked for orthogonality over every pair of rows.
Named alongside it
The objects these essays reach for when they reach for this one.
DeterminantEigenvalueEigenvectorMatrixCharacter tableCharacteristic polynomialConjugacy classDefective matrixDifferential equatione, the numberGroup representationInvariant direction