In a homework assignment I'm working on, a particular matrix norm $\left \| \cdot \right \|$ is defined as follows:
$\left \| A \right \| = \max\limits_{j} \left \| a_j \right \|_2$
Where $A\in\mathbb{C}^{m\times m}$, $a_j$ are the column vectors of $A$, and $\left \| \cdot \right \|_2$ is the euclidean distance vector norm.
The assignment then states that this matrix norm can be equivalently defined as follows:
$\left \| A \right \| = \max\limits_{x} \left \| Ax \right \|_2$
where the maximum is taken not over all unit vectors in $\mathbb{C}^m$, but just over the canonical basis vectors of $\mathbb{C}^m$
I don't understand this equivalent definition. I thought that "all unit vectors in $\mathbb{C}^m$" was the same thing as "the canonical basis vectors of $\mathbb{C}^m$". What's the difference?
Best Answer
The canonical basis is $$\{(1,0,0,\dots,0),(0,1,0,\dots,0),\dots,(0,0,0,\dots,0,1)\}$$ Obviously, there any many more unit vectors.