Miscellaneous

• Row rank is always equal to column rank.

• Any matrix mulitiply by a non-singular matrix, its rank will not change.

• Variance-covairance matrix is non-negtive. Positive deifinite => full rank, but full rank not => positive definite. (see example in other notes)

• A=MB, if M is invertible, $B=M^{-1}A$, which means $R(A)\subset R(B)$ and $R(B)\subset R(A)$. Thus, $R(A)=R(B)$.

• If A, B have the same row space=>rank(A)=rank(B). Because $A=K_{1}B$ and $B=K_{2}A$ => $rank(A)\le rank(B)$ and $rank(B)≤rank(A)$