Concept

Singular matrix

A matrix that collapses space onto something smaller, so that it cannot be undone and its determinant is zero. It has a non-trivial kernel, so the equation it defines either has no solution or a whole family of them.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

A linear map redrawing the plane. The integer grid before and after a linear transformation; the shaded unit square becomes a parallelogram whose area is the determinant.

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.

algebra · Linear maps

The unit square, mapped: area × 5. The unit square and the parallelogram it becomes under a linear map, with the area of that parallelogram computed from its own corners and set against ad − bc.

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.

algebra · Determinant

A whole line arrives at the origin. A linear map whose determinant is zero, drawn before and after. One line of the plane is sent to the origin and the whole plane is sent onto another line; the dimension lost and the dimension kept add to two.

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.

algebra · Linear maps

Named alongside it

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

Basis Determinant Matrix Area Orientation Rank Shear Dimension Eigenvalue Image Invariant direction Kernel

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