![]() ![]() On the other hand, since $A,B$ being nonsingular gives us that $AB$ is nonsingular, then we also have that 2) if $AB$ is singular, then at least one of $A$ and $B$ must be singular as well. van der pol circuit SINGULAR PERTURBATIONS FOR THE FORCED VAN. Notice, by the way, that we also showed that 1) $AB$ is singular if $B$ is the one assumed singular. The Matrix form of the Van der Pol equation - MathOverflow. But now we have that $(AB)a=A(Ba)=Ab=0$, and we conclude (again) that $AB$ is singular. Now, use that $B$ is nonsingular to find $a$ such that $Ba=b$. ![]() If, on the other hand, $B$ is nonsingular, use that $A$ is singular to find $b\ne 0$ such that $Ab=0$. This technique was reinvented several times. ) This strategy is particularly advantageous if A is diagonal and D CA 1 B (the Schur complement of A) is a small matrix, since they are the only matrices requiring inversion. Furthermore, A and D CA 1 B must be nonsingular. (A must be square, so that it can be inverted. If $B$ is also singular, then for some $x\ne 0$ we have $Bx=0$, but then $(AB)x=A(Bx)=A0=0$, and we conclude that $AB$ is also singular. 1) where A, B, C and D are matrix sub-blocks of arbitrary size. That $C$ is nonsingular, on the other hand, gives us in particular that the column space of $C$ has full rank, that is, for any vector $b$ there is a vector $a$ such that $Ca=b$. In electrical engineering, modified nodal analysis or MNA is an extension of nodal analysis which not only determines the circuits node voltages but also. One approach is this: That a matrix $C$ is singular gives us in particular that its null space is non-trivial, that is, for some vector $x\ne0$ we have $Cx=0$. ![]()
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