Determinant
Named by 28 essays across 5 fields — each of them below, with the objects they name alongside it.
A matrix is a picture of what happens to the grid
Four numbers in a box is not an object anyone has intuitions about. The same four numbers, shown as an instruction for redrawing the plane, are.
The directions a map leaves alone
Almost every arrow comes out of a transformation pointing somewhere else. A few come out pointing exactly where they went in, only longer or shorter. Those few decide nearly everything the map does.
The number that says how much room is left
A linear map takes the unit square to a parallelogram. The area of that parallelogram is one number, it is computable from the four entries of the matrix, and almost everything the determinant is used for is a restatement of that sentence.
What a map throws away
A linear map redraws the grid, and the determinant measures how much it stretches area. When that measurement comes out zero the map has flattened the plane onto a line — and the question worth asking is not how much was lost but how much survived, because the two always add to what there was.
The flat map that fits closest
A derivative is usually met as a number, which works because a line through a point is described by one. In more than one dimension the object that plays the same role is a linear map, and the number was always a one-by-one instance of it.
A shared root, found without finding it
Two polynomials have a root in common exactly when one determinant built from their coefficients is zero. No root is computed, nothing is approximated, and the same construction turns two equations in two unknowns into one equation in one.
The crossings that will not come out even
Draw a rearrangement as strings from one row of pegs to another and count where they cross. The count depends on how the strings are drawn; whether it is odd or even does not, and that single bit is what makes determinants exist and a sliding puzzle unsolvable.
![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.
![What the map does to a circle. The unit circle with two perpendicular directions marked, and its image under [1.6, 1.2, −0.4, 1.1] — an ellipse whose axes are the images of those two directions, of lengths 2.04 and 1.10.](/f/what-the-map-does-to-a-circle-29ac1dac.svg)
What a map does to a circle
Every linear map sends the unit circle to an ellipse. Two perpendicular directions go to two perpendicular directions, whatever the map is — even a map with no invariant direction at all, and even one that is not square.
![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.
Orientation is a sign
Carry a pair of arrows once round a loop and compare them with the pair they started as. The comparison is a determinant, its sign is the whole of the answer, and no room, no side and no normal vector appears anywhere in the statement.
The only function that behaves like a volume
Ask for a function of the columns of a matrix that scales when a column scales, vanishes when two columns agree, and gives one on the identity. Three conditions, and there is exactly one such function in every dimension.
One point in every big enough shape
A determinant measures a lattice, not the basis that happened to describe it — and that measurement is an exchange rate. Any symmetric convex region with more than four times that area has to swallow a lattice point.
A determinant that counts trees
Write down a graph's Laplacian, strike out one row and its column, take the determinant. The answer is the number of spanning trees — and the minus signs in the determinant are what cancel every subset of edges that is not one.
The same sum without its minus signs
Delete the signs from the determinant's sum over permutations and what is left counts things directly rather than by cancellation. It is a better count and a far worse object — because the cancellation was what made the determinant computable.
Two matrices that generate the tree
A node of the Stern–Brocot tree is not really a fraction — it is the pair of fractions it lies between. Written as the columns of a matrix, the two turns of the tree become two multiplications, and the determinant that kept everything in lowest terms becomes a property of a product.
A plane disguised as an arrow
The cross product of two arrows is an arrow perpendicular to both, as long as the area of their parallelogram. Reflect everything in a mirror and it points the wrong way, because it was never an arrow: it is a plane, written as the one direction a plane in three dimensions leaves over.
The arcs a line crosses on its way to a number
Draw a semicircle over every pair of neighbouring fractions and the half-plane above the number line is cut into curved triangles that never overlap. A straight line dropped towards any number crosses those arcs one after another, and the arcs it crosses, and the side it leaves each triangle by, are exactly the steps of the Stern–Brocot descent towards that number.
A centre is three weights
Write each classical centre as a weighted average of the corners and a coincidence becomes a determinant. The Euler line is then one number rather than a construction, the whole catalogue becomes mechanical, and the reason one centre is missing from it is visible in the weights.
A polynomial behind the colourings
The figure-eight knot and the cinquefoil both have determinant five, so they admit exactly the same colourings, and every counting argument treats them as one. Put a variable where the colouring rule has a two and the determinant becomes a polynomial — and the two knots come apart.
![The integers of ℚ(√5), with ℤ[√5] inside them. Points a + bφ plotted against their conjugates for small whole a and b, with the index-two sublattice ℤ[√5] filled and the basic cells of both lattices shaded, of areas √5 and 2√5.](/f/the-integers-of-q-5-with-z-5-inside-them-a4f067e7.svg)
The integers a field contains
Inside the field of numbers a + b√5, the obvious integers are those with whole a and b. They are not all of them: the golden ratio has a one-half in it and satisfies x² = x + 1, a monic equation with whole coefficients, exactly as an integer should. The right integers form a lattice twice as dense as the obvious one — and a whole-number matrix proves they are closed under addition.
Units that form a lattice
In the whole numbers only 1 and −1 have whole-number reciprocals. In the integers of a bigger field there can be infinitely many such units, and they are not scattered: take logarithms of their sizes under each way of placing the field in the real or complex numbers, and the units land exactly on a lattice. How many dimensions that lattice has is a count of those placements, and the area of its cell is a number no formula gives.
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.
The biggest box built from signs
Fill a square table with plus and minus ones and ask how large its determinant can be. The columns all have the same length, so the answer is a box with fixed edges — largest when every corner is square, which is possible only when the size is a multiple of four.
The count a fold cannot change
A curved map can fold the plane over itself, so that one point has three preimages and its neighbour has one. Count each preimage with the sign of the determinant there and the jump disappears — the signed count is the same everywhere, and it is a whole number.
Every section of the band must vanish
Read the Möbius band as a line standing over each point of a circle, and choose a point on each line continuously: a section. On a cylinder a section can stay away from zero all the way round. On the band it cannot — every section crosses zero an odd number of times — and that single fact is what orientability means for a bundle. There are exactly two line bundles over a circle, told apart by one sign, and two Möbius bands added together make the plain cylinder's twin.
An area measured by walking round it
A surveyor's instrument from 1854 measures the area of any shape on a map by having its pointer steered once round the boundary; a small wheel rolls and slides, and its reading, times the length of one arm, is the area. Nothing touches the inside. The reason is that area can be written as an integral over the boundary — Green's theorem — and the instrument is that integral built in brass. Walk a curve that crosses itself and the same integral counts some regions twice.
Named alongside it
The objects these essays reach for when they reach for this one.
MatrixOrientationBasisEigenvalueAreaInvariantCounting argumentEigenvectorOrthogonalityShearTraceApproximation