Linearity, Compatibility with the Matrix Product, and a Norm Bound for the Matrix-Vector Product

Let $m$, $n$ and $p$ be natural numbers and let $\mathbb{R}$ be the real numbers, an ordered field with additive identity $0$ and order $\le$; write $|t|$ for the absolute value of $t\in\mathbb{R}$. Let $A$ be a real matrix with $m$ rows and $n$ columns, its entry in row $k$ and column $i$ written $A_{ki}$, and let $B$ be a real matrix with $p$ rows and $m$ columns, its entry in row $\alpha$ and column $j$ written $B_{\alpha j}$. Let $v=(v_1,\dots,v_n)$ and $w$ be points of Euclidean space $\mathbb{R}^n$, and let $\mu\in\mathbb{R}$.

Write $Av$ for the matrix-vector product, $BA$ for the product of real matrices, and $\lVert\,\cdot\,\rVert$ for the Euclidean norm, used on $\mathbb{R}^n$ and on $\mathbb{R}^m$ alike. On $\mathbb{R}^n$ and $\mathbb{R}^m$, regarded as real vector spaces by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, write $v+w$ for the sum of points, $\mu v$ for the scalar multiple, $v-w$ for the difference of points, and $0_{\mathbb{R}^n}$ for the origin. Sums below are finite sums in $\mathbb{R}$, and index ranges such as $1\le i\le n$ use the order on the natural numbers.

Then the following hold.

1. (Linearity in the vector)

$$A(v+w)=Av+Aw,\qquad A(v-w)=Av-Aw,\qquad A(\mu v)=\mu\,(Av),\qquad A\,0_{\mathbb{R}^n}=0_{\mathbb{R}^m}.$$

2. (Compatibility with the matrix product)

$$(BA)v=B(Av).$$

3. (Norm bound) For every natural number $k$ with $1\le k\le m$ put

$$c_k=\sum_{i=1}^{n}|A_{ki}| ,$$

and put $C=\lVert(c_1,\dots,c_m)\rVert$. Then $0\le C$ and

$$\lVert Av\rVert\le C\,\lVert v\rVert .$$

In particular there exists a real number $C$ with $0\le C$ such that $\lVert Av\rVert\le C\,\lVert v\rVert$ for every $v\in\mathbb{R}^n$.