Cauchy-Schwarz inequality of determinants:


for $A\_{n\times k}$, $B\_{n\times k}$, and $B'B$ non-singular, we have 


$|A'B|^2\leq |A'A||B'B|$


I was wondering what's the sufficient and necessary conditions for the equality to hold.


I know a sufficient condition:


when $\exists C\_{k\times k}$, such that $A=BC$, then the equality holds.


Is this also a necessary condition?


Thanks.



Sufficient and necessary conditions on equality of Cauchy-Schwarz inequality of determinants
The standard argument goes as follows: Let $M=B'A(A'A)^{-1}A'B$ and $N=B'(I-A(A'A)^{-1}A')B$. Then it is easy to check that $M$ and $N$ are positive definite and that the Cauchy-Schwarz inequality is equivalent to the following (true) inequality
$$
|M+N| \ge |M|
$$which has equality if and only if $N$ is the zero matrix, i.e., when $(I-A(A'A)^{-1}A')B=0$, or $B=AC$ where $C=(A'A)^{-1}A'B$. This shows that your inequality is an equality if and only if $B=AC$ where $C$ is invertible.


