Is there a name for $\max \| A x \|$ for all $\|x\|=1?$

Question

Is there a name for $\max \| A x \|$ for all $\|x\|=1?$

($A$ is a matrix, and $x$ is a vector)

One might be inclined to guess that it's the spectral radius of $A$, but that's not true. I'm wondering if there is a special term for this property of $A$.

Answer 1

This is the norm of a linear map or the operator norm. It checks "how much does this matrix stretch values."

It is a theorem that $$\mathrm{Sup}_{x \in X}\frac{\|Ax\|}{\|x\|}=\mathrm{Sup}_{\|x\|=1}\|Ax\|.$$

Answer 2

It is called the induced matrix norm. If the underlying vector norm is the Euclidean one, then it is also called the largest singular value.

Answer 3

It goes by the name of operator norm. It's equal to $\sqrt{\rho(A^HA)}$ when the norms are the usual euclidean ones.

Answer 4

Can be called a subordinate norm.

Answer 5

As already mentioned, it is called the operator norm (more precisely the operator norm of the linear transformation induced by the map $x \to Ax$).

I will add that it is frequently used in real analysis to "bound things". In particular, the operator norm satisfies the following (extremely) useful property:

$$\Vert Ax \Vert \leq \Vert A \Vert \Vert x \Vert \quad \text{for all }x$$

Answer 6

It looks like it's also called the natural norm.