Transposition and Matrix-Matrix Products

The transpose of a m by n matrix A is a n by m matrix A^T such that A^T[i,j] = A[j,i]. We've exchanged rows and columns.

The product of two matrices A (m by n) and B (n by p) is a matrix C (m by p) where $C[i,j] = \sum_k A[i,k] B[k,j]$.Note that this is most often computed using a simple triply nested loop.

Matrix products come up a lot, too.

• If A is the matrix of floor-floor transition probabilities (A[i,j] is the probability that the elevator will next be on floor j given that it is currently on floor i), then A A = A² is the matrix of two-step transition probabilities (the i,j entry is the probability that the elevator will be on floor j two stops from now given that it is currently on floor i).