A 2×2 matrix holds just four numbers, and those four numbers say nothing except where the two unit arrows î and ĵ end up. The whole coordinate grid follows them: the determinant is the factor by which every cell’s area grows, a negative determinant means the plane has been mirrored, and a determinant of zero collapses the entire plane onto a single line, leaving the map non-invertible. Eigenvectors are the rare directions that are not swung aside, which is why a 30-degree rotation has none of them among the reals.
A 2×2 matrix holds just four numbers, and those four numbers say nothing except where the two unit arrows î and ĵ end up. The whole coordinate grid follows them: the determinant is the factor by which every cell’s area grows, a negative determinant means the plane has been mirrored, and a determinant of zero collapses the entire plane onto a single line, leaving the map non-invertible. Eigenvectors are the rare directions that are not swung aside, which is why a 30-degree rotation has none of them among the reals.
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