Let $A\in M_{3\times 3}$ and $b, c\in\mathbb{R}^3$ be two nonnull vectors with $|b|\le 1$. Consider the dot product
$$ Mb\cdot c.$$
I would like to know if my following argument is valid.
As $Mb$ is a vector, then $$ M b\cdot c \le |M\cdot b| |c|\le \|M\| |c|,$$
where $\|M\|$ denotes the matrix-norm of $M$ and I used the assumption $|b|\le 1$.
Do you think my argument is valid?
Thank you in advance.
The first inequality $Mb\cdot c\leq|Mb||c|$ is known as Cauchy Schwartz and the second inequality $|Mb||c|\leq\|M\||b||c|$ follows from the definition of operator norm so your argument is valid, provided you use the operator norm as your matrix norm. (Turns our there are many many matrix norms).