Concept

Similarity

The relation between two square matrices that are the same linear map written in different bases, one equal to P⁻¹AP for the other. Similar matrices share eigenvalues, trace and determinant, because those belong to the map and not to the coordinates used to write it down.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The same map, written in the basis of its own eigenvectors. Three panels: the map [2, 1, 1, 2] on the standard grid, the diagonal stretch by 3 and 1 it becomes on the eigenvector grid, and the two put back together.

The same map in a better basis

Measured along its own invariant directions, a linear map stops shearing and becomes two independent stretches. Nothing about the map has changed; the grid it is described against has.

algebra · Eigenvectors

Discs from the rows, and the eigenvalues inside them. The complex plane with one disc per row of a 3×3 matrix — centred on its diagonal entry, with radius the sum of the sizes of the rest of the row — and the 3 eigenvalues marked: 4.14, −5.09, −1.06.

Discs that fence in the eigenvalues

Draw one disc for each row of a square matrix, centred on the diagonal entry, with a radius equal to the sum of the sizes of everything else in that row. Every eigenvalue lies inside one of the discs, and a group of discs set apart from the rest holds exactly as many eigenvalues as it has discs. Nothing is solved to find them.

algebra · Eigenvectors

Named alongside it

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

Eigenvalue Matrix Basis Change of basis Complex numbers Continuity Convergence rate Defective matrix Diagonal dominance Diagonalisation Eigenvector Estimate

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