eigenbasis

An eigenbasis is a basis for the whole space made entirely out of eigenvectors of a given matrix. Instead of describing vectors with the usual axes, you describe them along the matrix's own natural directions, the ones it merely stretches.

This is the right coordinate system for understanding the transformation, because in it the matrix becomes diagonal. Every basis vector just gets scaled by its own eigenvalue, with no mixing between directions. The complicated bundle of stretching and rotating dissolves into a list of independent stretches, one per axis.

Example: A = [[2,0],[0,3]] already acts on the standard basis (1,0) and (0,1) as pure stretches by 2 and 3, so those two vectors form an eigenbasis and the matrix is already diagonal. For a less tidy matrix, you first solve for its eigenvectors and use them as the new axes.