# Matrix Norm

Matrix norm is norm that defined on matrixes. It satisfies the mathematical properties of norms.

## Definitions
### operator matrix norm (作用素ノルム)
Matrix norm $|A|_{p}$ is defined by
\begin{eqnarray}
|A|_ {p} = \sup_{x\neq 0} \frac{|Ax|_ p}{|x|_ p}, \tag{1}
\end{eqnarray}
where $p$ represents for p-norms. Operator norm is also called induced norm (誘導ノルム).

We can also use $\max$ insted of $\sup$ when defining norm. $x$ is not a fixed vector, but we vary them to find the maximum of RHS of eq(1) to calculate matrix norm.

For simplication denominator of RHS of eq(1) is replaced to 1, which means $x$ is a unit vector:
\begin{eqnarray}
|A|_ {p} = \max_{|x|=1} |Ax|_ p. \tag{2}
\end{eqnarray}

#### $p=1$ and $p=\infty$
\begin{eqnarray}
Ax = \sum_{j}^{} A_{ij} x_j
\end{eqnarray}

$p=1$
\begin{eqnarray}
|A|_ {1} =&& \max_{|x|_ 1=1} |Ax|_ 1\\
=&& \max_{|x|_ 1=1} \sum_{ij}^{} |A_{ij} x_j|\\
=&& \sum_{i}^{} |A_{ij_{\max}}|
\end{eqnarray}
where $j_{\max}$ is the index of column vector of $A$ that the sum of elelmnts are at maximum. Thus, the 1-norm of the matrix is the maximum of sum of absolute values of elements of the column vectors.

$p=\infty$
\begin{eqnarray}
|A|_ {\infty} =&& \sup_{|x|_ \infty=1} |Ax|_ \infty\\
=&& \max_{|x|_ \infty=1} \max_i \left(\left|\sum_{j}^{}A_{ij} x_j\right|\right)\\
=&& \max\left(\sum_{j}^{}|A_{i_\max j}|\right)
\end{eqnarray}
Thus the $\infty$ norm of the matrix is the maximum of sum of the absolute value of elements the row vectors.

#### $p=2$
\begin{eqnarray}
|A|_ {2} =&& \max_{|x|_ 2=1} |Ax|_ 2\\
=&& \max_{|x|_ 2=1} \left( \sqrt{x^TA^T Ax}\right)\\
=&& \max_{|x|_ 2=1} \max_i {\sqrt{\sum_{i}^{} \lambda_i(A^TA)|x_i|^2}}\\
=&& \sqrt{\lambda_{i_\max}(A^TA)}
\end{eqnarray}
where $\lambda_{i}(A^TA)$ is i-th eigenvalue of $A^TA$.
The 2-norm of the vector corresponds to the maximum of the singular value of $A$. This is also called the **spectral norms**.

### Schatten norm (To be editted)

### "Entry-wise" matrix norms (To be editted)

## Properties as norms (To be editted)
In this section, we will check that the above norms satisfies the properties of norms.

## References
https://en.wikipedia.org/wiki/Matrix_norm#%22Entrywise%22_matrix_norms
https://mathwords.net/matrixpnorm