Let $a$, $c$, $d$, $v \in \mathbb{R}^n$ are vectors, and $A, B \in \mathbb{R}^{n \times n}$ are matrices. Suppose that $ v = Ac-d $, and $a = ABc- \|B\| d$ where $\| B \|$ is the maximum value of the norms of the eigenvalues of $B$.


Is it true that $\| a \| \leq \|B\| \| v\|$?



Upper-bounds for a vector equation
No, it is not true in general.


Example: $A:=I$ (identity). $B:=-I$. $c\neq 0$ arbitrary. $d:=c$.


Then $v=Ac-d=0$ and $a=ABc-\lVert B\rVert d=-c-1d=-2c\neq 0$. Then $\lVert a\rVert > 0$ and $\lVert B\rVert\lVert v\rVert=0$.


