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Concept

Matrix exponential

The power series 1 + A + A²/2 + A³/6 + …, applied to a square matrix A, which always converges. It solves the linear differential equation with matrix A, and unlike the logarithm it can be applied to every square matrix.

Named by 3 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.

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.

analysis · The exponential The trace–determinant plane and the flows it sorts. The plane of trace against determinant, divided by the horizontal axis and the parabola tr² = 4 det into saddle, node, spiral and centre regions, with 6 matrices marked and their phase portraits drawn alongside.

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.

analysis · The exponential Which real 2 × 2 matrices have a real logarithm, read off their eigenvalues. Eigenvalues of 7 matrices plotted in the complex plane with the negative real axis emphasised, beside a table saying for each whether a real logarithm exists and why: A yes, B yes, C yes, D no, E no, F yes, G no.

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.

analysis · The exponential

Named alongside it

The objects these essays reach for when they reach for this one.

DeterminantEigenvalueTraceDifferential equationPower seriesRotatione, the numberEigenvectorLogarithmMatrixPhase portraitStability

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