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Concept

Shear — where it appears

A map that slides each line parallel to a fixed direction by an amount depending on its distance, leaving area unchanged. It is what Euclid's proof of the Pythagorean theorem uses, since sliding a parallelogram along its base leaves its area alone.

Named by 5 essays across 2 fields — 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 directions the map leaves alone. Unit vectors and their images under the map. On the two marked lines the image points the same way as the original, stretched by 3.00 and 1.00.

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.

algebra · Eigenvectors Euclid's proof, without moving anything. The square on a leg and its share of the square on the hypotenuse are each exactly twice the same triangle, so they are equal. Nothing in the figure is cut or rearranged; the triangle is only looked at from the other side.

Euclid proves it without moving anything

The rearrangement proof cuts and slides. Euclid's does neither — it shows that a square and a rectangle are each exactly twice the same triangle, seen from opposite sides, and that is harder to hold in the head for a reason worth understanding.

geometry · Pythagoras 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 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.

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.

algebra · Eigenvectors

Named alongside it

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

DeterminantMatrixAreaBasisEigenvalueInvariant directionCharacteristic polynomialEigenvectorOrientationSingular matrixAltitudeCongruence

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