For two finite-dimensional real vector spaces $E_1,E_2$, define an linear operator $A:E_1\to E_2^*$. Its adjoint operator is defined by $A^*:E_2\to E_1^*$ its adjoint operator, i.e. $$\langle Ax,u \rangle_{E_2} = \langle A^*u,x \rangle_{E_1} \quad \forall x\in E_1,\forall u\in E_2.$$
The norm of $A$ is given by: $$\|{A}\| = \max_{x\in E_1,u\in E_2} \left\{\langle Ax,u \rangle_{E_2}: \|x\|_{E_1}=\|u\|_{E_2}=1\right\}.$$
Now I want to compute the norm of $A$ when $E_1=\mathbb{R}^n$ and $E_2=\mathbb{R}^m$.
$A$ mapping $\mathbb{R}^n$ to $\mathbb{R}^{m*}$ is represented by an $m\times n$ matrix $\mathbf{A}$. The adjoint operator $A^*$ mapping $\mathbb{R}^m$ to $\mathbb{R}^{n*}$ is represented by $\mathbf{A}^T$.
I have found the following results:
- If both $E_1$ and $E_2$ have the $L^1$ norm: $\|{A}\|=\max_{i,j}|a_{ij}|$.
- If both $E_1$ and $E_2$ have the $L^2$ norm: $\|{A}\|= \sqrt{\lambda_{\mathrm{max}}} \mathbf{A}^T \mathbf{A}$ where $\lambda_{\mathrm{max}}$ is the maximum eigenvalue of $\mathbf{A}$.
- If $E_1$ has the $L^2$ and $E_2$ has $L^1$ norm: $\|{A}\|= \max_{1\le i \le m} \sqrt{\sum_{j=1}^n a_{ij}^2}$.
- If $E_1$ has the $L^1$ and $E_2$ has $L^2$ norm: no idea.
Hope somebody can help to complete the last case. Thank you in advance.
The induced norm of the matrix $A$ as a map from $(\mathbb R^n , \Vert \cdot \Vert_1)$ to $(\mathbb R^n, \Vert \cdot \Vert_2)$ is given by $$ \max_{j=1, \dots ,n} \Vert A^j \Vert_2, $$ where $A^j$ is the $j^{th}$ column of $A$.
You can find this as equation (2c) in
"On the Calculation of the $l_2 \to l_1$ Induced Matrix Norm"
Konstantinos Drakakis and Barak A. Pearlmutter,
International Journal of Algebra, Vol. 3, 2009, no. 5, 231 - 240
This is proved in that paper (among other results) but they say that the specific result above was previously shown in
N. Higham. Estimating the matrix p-norm. Numerische Mathematik, 62(1):539–55, Dec. 1992. doi: 10.1007/BF01396242.
In general, one should not expect there to be a short, explicit formula for the induced matrix norm. For more information on that, I direct you to Is the problem of calculating the induced norm *difficult*?