Similar Matrices

Linear Algebra (Matrices) · Axiom Academy

Two matrices are similar when B = P^ -1 AP — the same transformation seen in a different basis, sharing every invariant. Square matrices A and B are similar (written ) when some invertible matrix P — the change-of-basis matrix — satisfies B = P^ -1 AP . Watch the sandwich assemble around A and resolve to a concrete B . B is the same map in the basis given by the columns of P Because A and B are the same transformation, any quantity that belongs to the transformation — not to the basis — must come out identical. Watch the four invariants get computed on both sides and meet in the middle on one shared value. 3. Reading P^ -1 AP as a Round Trip Apply B to a vector written in the new basis and follow the three moves: P sends it to standard coordinates, A transforms it there, and P^ -1 brings the result home. Watch one vector make the whole loop — landing exactly where B alone would send it. Rebuild the vector from the new basis vectors (the columns of P ). Do the actual transformation where A lives: the standard basis. Re-express the transformed vector in the new basis coordinates. B = P^ -1 AP collapses the round trip into a single matrix. You've seen similarity as one transformation wearing two matrices — and why every invariant carries across unchanged. Scroll up to revisit any step.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.