Is there any relation between the matrix' operator norm and its column vectors?
For instance if I have a matrix $A$ that is $3 \times 3$ and formed column-wise by three vectors $v_1, v_2, v_3$ of lengths $l_1, l_2, l_3$, can I compute or give a bound for the norm $\|A\|$?
What can be said about the norm of the inverse if $v_1, v_2, v_3$ are all linearly independent?
Best Answer
Let $A=\begin{pmatrix} \vec{v_1}& \vec{v_2} & \vec{v_3} \end{pmatrix}$ and consider a column vector $\vec{x} = \begin{pmatrix} x \\ y \\ z\end{pmatrix}$. Then $$ A\vec x=\begin{pmatrix} \vec{v_1}& \vec{v_2}& \vec{v_3} \end{pmatrix} \begin{pmatrix} x \\ y \\ z\end{pmatrix}= x \vec{v_1} + y\vec{v_2} + z\vec{v_3}. $$
By triangle inequality,
$$ ||A\vec x|| \leq |x| ||\vec{v_1} || + |y| ||\vec{v_2}|| + |z| ||\vec{v_3}||. $$
By Cauchy-Schwarz inequality, $$ |x| ||\vec{v_1} || + |y| ||\vec{v_2}|| + |z| ||\vec{v_3}|| \leq (x^2+y^2+z^2)^{\frac12} (||\vec{v_1}||^2 + ||\vec{v_2}||^2+||\vec{v_3}||^2)^{\frac12}. $$ Therefore, we have $$ ||A ||\leq (||\vec{v_1}||^2 + ||\vec{v_2}||^2+||\vec{v_3}||^2)^{\frac12}=(l_1^2+l_2^2+l_3^2)^{\frac12}. $$