0

I have a matrix equation of the form
$$A = (B + C)^{-1} D$$ Can this equation be interpreted as $(B + C)$ inversely proportional to A?
ie. Suppose C is just an identity matrix multiplied by some scalar, does increasing the scalar value decrease the values of $A$ if we keep the rest of the terms constant?

Update1:
I tried increasing the scalar value, this actually decreases the values in A. However, I cannot find any statements/sources regarding this on the Internet.

Coolboy
  • 35

1 Answers1

0

Yes, you can interpret it in this manner. One way to formalize it would be to consider the right hand side as a function of the matrix entries in $B,C,D$. To see the effect of modifying any particular matrix entry, you can take the matrix derivative with respect to that entry. The calculation is similar to what happens when $B,C,D$ are $1\times 1$ matrices.

pre-kidney
  • 30,223
  • Thanks, based on your suggestion of differentiation I tried differentiating the equation. See here. I created a separate question since the scope of this question was limited to interpreting matrix inversion – Coolboy Feb 27 '22 at 03:21