Matrix Multiplication and Normalization

Question

$A$ is a $J \times K$ rectangular matrix.

$B$ is a $K \times L$ rectangular matrix.

$B$ is column-normalized: $\forall l: 1 = \sum_{k = 1}^{K}B_{kl}$

$AB$ is column-normalized: $\forall l: 1 = \sum_{j = 1}^{J}\sum_{k=1}^{K}A_{jk}B_{kl}$

Does this mean $A$ must also be column-normalized: $\forall k: 1 = \sum_{j = 1}^{J}A_{jk}$ ?

Thanks in advance.

Answer 1

Generally not true. Even for square matrices, it requires orthogonal matrices. AB times B inverse is A. Product of normalized matrices is a normalized matrix.