834: Matrices Multiplications Map Is Continuous

<The previous article in this series | The table of contents of this series | The next article in this series>

description/proof of that matrices multiplications map is continuous

---

Topics

About: topological space

---

The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Proof

---

Starting Context

• The reader knows a definition of %field name% matrices space.
• The reader knows a definition of Euclidean topology.
• The reader knows a definition of product topology.
• The reader knows a definition of continuous map.

---

Target Context

• The reader will have a description and a proof of the proposition that any matrices multiplications map is continuous.

---

Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

---

Main Body

---

1: Structured Description

Here is the rules of Structured Description.

Entities:
$M(d_{1}x{d}_2,\mathbb{R})$: $=\{\text{ the }{d}_{1}x{d}_2\text{ real matrices }\}$ with the topology of the Euclidean topological space, ${\mathbb{R}}^{d_1{d}_2}$
$M(d_{2}x{d}_3,\mathbb{R})$: $=\{\text{ the }{d}_{2}x{d}_3\text{ real matrices }\}$ with the topology of the Euclidean topological space, ${\mathbb{R}}^{d_2{d}_3}$
$M(d_{1}x{d}_3,\mathbb{R})$: $=\{\text{ the }{d}_{1}x{d}_3\text{ real matrices }\}$ with the topology of the Euclidean topological space, ${\mathbb{R}}^{d_1{d}_3}$
$f$: $:M(d_{1}x{d}_2,\mathbb{R})\times M(d_{2}x{d}_3,\mathbb{R})\to M(d_{1}x{d}_3,\mathbb{R}),(M,N)\mapsto MN$
//

Statements:
$f\in \{\text{ the continuous maps }\}$
//

---

2: Proof

Whole Strategy: Step 1: take each $(M,N)\in M(d_{1}x{d}_2,\mathbb{R})\times M(d_{2}x{d}_3,\mathbb{R})$ and each open cube around $MN$, $C_{MN,ϵ}$; Step 2: take an open neighborhood of $(M,N)$, $C_{M,\delta }\times C_{N,\delta }$, and choose $\delta$ to satisfy $f(C_{M,\delta }\times C_{N,\delta })\subseteq C_{MN,ϵ}$.

Step 1:

Let $(M,N)\in M(d_{1}x{d}_2,\mathbb{R})\times M(d_{2}x{d}_3,\mathbb{R})$ be any.

Let $C_{MN,ϵ}$ be any open cube around $MN$.

Step 2:

Let us take an open neighborhood of $(M,N)$, $C_{M,\delta }\times C_{N,\delta }$, where $C_{M,\delta }$ and $C_{N,\delta }$ are the open cubes: we are going to choose $\delta$ to satisfy $f(C_{M,\delta }\times C_{N,\delta })\subseteq C_{MN,ϵ}$.

$(MN{)}_k^j=M_l^j{N}_k^l$.

Let $\bar{M}$ be the maximum absolute component of $M$; let $\bar{N}$ be the maximum absolute component of $N$.

As $M_l^j$ becomes $M_l^j+{\lambda }_l^j$ and $N_k^l$ becomes $N_k^l+{\tau }_k^l$, $M_l^j{N}_k^l$ becomes $(M_l^j+{\lambda }_l^j)(N_k^l+{\tau }_k^l)=M_l^j{N}_k^l+M_l^j{\tau }_k^l+{\lambda }_l^j{N}_k^l+{\lambda }_l^j{\tau }_k^l$.

Its absolute difference from $M_l^j{N}_k^l$ is $|M_l^j{\tau }_k^l+{\lambda }_l^j{N}_k^l+{\lambda }_l^j{\tau }_k^l|\le |M_l^j||{\tau }_k^l|+|{\lambda }_l^j||N_k^l|+|{\lambda }_l^j||{\tau }_k^l|\le \sum _l\bar{M}\delta +\delta \sum _l|\bar{N}|+\sum _l{\delta }^2=d_2\bar{M}\delta +\delta d_2|\bar{N}|+d_2{\delta }^2$.

It is obvious that $\delta$ can be chosen small enough such that $d_2\bar{M}\delta +\delta d_2|\bar{N}|+d_2{\delta }^2<ϵ$.

That means that $f(C_{M,\delta }\times C_{N,\delta })\subseteq C_{MN,ϵ}$.

---

References

<The previous article in this series | The table of contents of this series | The next article in this series>