Matrix exponential
Named by 5 essays across one field — each of them below, with the objects they name alongside it.
![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.
Two numbers decide the flow
A linear system in the plane has four coefficients, and what its solutions do forever afterwards — spiral in, race out, swing round, or split along two lines — is decided by two of the numbers made from them. The plane of trace against determinant is a complete map of the possibilities, and the only places it cannot decide are the lines where it changes its mind.
The matrix that has no logarithm
Every square matrix has an exponential, and a matrix exponential is always invertible. The converse fails, and it fails in a way that can be read straight off the eigenvalues — a real matrix with eigenvalues −1 and −2 is no exponential at all, while minus the identity is the exponential of a whole continuum of matrices that do not even commute with each other.
When a table of moves came from steady rates
A process that jumps between states at constant rates, watched once a year, produces a table of yearly moves that is the exponential of its rates. Most tables that anyone could write down are not — a random three-state table is only about one time in twenty-three — and the few that are can have two different sets of rates behind them, but only once the process has forgotten where it started.
The route round that rates cannot avoid
A table of yearly moves between states came from steady rates only if it is the exponential of a rate matrix. With two states that is true of half of all tables, with three of one in twenty-four, with four of about one in two thousand, and of none of 400,000 tables drawn at random with five. Even a table that is nine parts staying put fails more and more often as states are added, and always for one reason: rates that carry 1 to 2 and 2 to 3 also carry 1 to 3, and a table whose direct move is rarer than that route round cannot have come from them.
Named alongside it
The objects these essays reach for when they reach for this one.
DeterminantEigenvalueLogarithmTraceDifferential equationMarkov chainPower seriesRotationTransition matrixe, the numberEigenvectorMatrix