# 「分割式」乘法

[線代](https://lochiwei.gitbook.io/math/linear) ⟩ [矩陣](https://lochiwei.gitbook.io/math/linear/matrix) ⟩ [運算](https://lochiwei.gitbook.io/math/linear/matrix/op) ⟩ [乘法](https://lochiwei.gitbook.io/math/linear/matrix/op/mult) ⟩ 「分割式」乘法

{% hint style="success" %}
在「[行向量 ⨉ 列向量](https://lochiwei.gitbook.io/math/linear/matrix/op/mult/outer-product)」的乘法中，我們提到：「[行向量](https://lochiwei.gitbook.io/math/linear/matrix/row-col)乘以[列向量](https://lochiwei.gitbook.io/math/linear/matrix/row-col)，會得到<mark style="color:yellow;">**一個表格**</mark>」，類似九九乘法表。事實上，我們可以進一步將這個表格進行「<mark style="color:yellow;">**分割**</mark>」，然後再填入相對應的數值，<mark style="color:yellow;">**結果一樣**</mark>。
{% endhint %}

{% tabs %}
{% tab title="⬆️ 需要" %}

* [行向量 ⨉ 列向量](https://lochiwei.gitbook.io/math/linear/matrix/op/mult/outer-product)
  {% endtab %}

{% tab title="⬇️ 應用" %}

* [「塊狀」乘法](https://lochiwei.gitbook.io/math/linear/matrix/op/mult/by-blocks)
  {% endtab %}

{% tab title="🗺️ 圖表" %} <img src="https://487293287-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M5-JmwIjjSzWft9bo8C%2Fuploads%2FQblIZ8kd5AVCDiKKQjhN%2Fmatrix.mult.by.split.svg?alt=media&#x26;token=b62312b8-6bb1-4463-a193-f82bb2c08376" alt="" class="gitbook-drawing">
{% endtab %}

{% tab title="👥 相關" %}

* [「分組式」乘法](https://lochiwei.gitbook.io/math/linear/matrix/op/mult/by-groups)
  {% endtab %}

{% tab title="📗 參考" %}

* [ ] Linear Algebra - A Modern Introduction, 3.1 Matrix Operations ⟩&#x20;
  * Partitioned Matrices
    {% endtab %}
    {% endtabs %}

在下圖中，我們將[行向量](https://lochiwei.gitbook.io/math/linear/matrix/row-col)  $${\color{blue}\mathbf{u}}$$ 分割成 $${\color{blue}\mathbf{u}}*{1}, {\color{blue}\mathbf{u}}*{2}$$，將[列向量](https://lochiwei.gitbook.io/math/linear/matrix/row-col) $${\color{red}\mathbf{v}}$$ 分割成 $${\color{red}\mathbf{v}}\_1 , {\color{red}\mathbf{v}}\_2$$：

<figure><img src="https://487293287-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M5-JmwIjjSzWft9bo8C%2Fuploads%2FiJZUvHiLEQfsxOATqZTp%2Fuv_split_table.png?alt=media&#x26;token=6e5d2b73-e1c8-45eb-891e-8da58118c92d" alt=""><figcaption><p>「分割式」乘法</p></figcaption></figure>

在上圖中，<mark style="color:yellow;">**分割表格**</mark>對填入數值的方式<mark style="color:yellow;">**沒有任何影響**</mark>，我們還是利用<mark style="color:yellow;">**類似填九九乘法表**</mark>的方式，在每個儲存格中填入該有的數值，結果當然會一樣。

有意思的是：

{% hint style="success" %}
$${\color{blue}\mathbf{u}}$$, $${\color{red}\mathbf{v}}$$ 兩向量的<mark style="color:yellow;">**分割方式各自獨立**</mark>，要<mark style="color:green;">**怎麼分割都可以**</mark>，這點跟[「分組式」乘法](https://lochiwei.gitbook.io/math/linear/matrix/op/mult/by-groups)不同。
{% endhint %}

這裡要注意幾點：

{% hint style="warning" %}

* 圖中的 $${\color{blue}\mathbf{u}}*{1}, {\color{blue}\mathbf{u}}*{2}, {\color{red}\mathbf{v}}*{1}, {\color{red}\mathbf{v}}*{2}$$ 都是用<mark style="color:yellow;">**大寫**</mark>，它們都是 [(行或列) 向量](https://lochiwei.gitbook.io/math/linear/matrix/row-col)，<mark style="color:red;">**不是純數**</mark>:exclamation:
* 這些向量也是[矩陣](https://lochiwei.gitbook.io/math/linear/matrix)，這種切割過的矩陣稱為「<mark style="color:yellow;">**子矩陣**</mark>」(submatrix)。
* $${\color{blue}\mathbf{u}}*{1} {\color{red}\mathbf{v}}*{1} \ \cdots \ {\color{blue}\mathbf{u}}*{2} {\color{red}\mathbf{v}}*{2}$$ 這些矩陣也都是<mark style="color:yellow;">**子矩陣**</mark>，<mark style="color:yellow;">**大小不盡相同**</mark>，不能當作純數對待:exclamation:
  {% endhint %}

我們將這種觀點整理成以下引理：

## 💍 引理 <a href="#lemma" id="lemma"></a>

假設我們將行向量  $${\color{blue}\mathbf{u}}$$ 分割成 $$\begin{bmatrix} {\color{blue}\mathbf{u}}\_1 \ {\color{blue}\mathbf{u}}\_2 \ \vdots \ {\color{blue}\mathbf{u}}\_m \end{bmatrix}$$，列向量 $${\color{red}\mathbf{v}}$$ 分割成 $$\begin{bmatrix} {\color{red}\mathbf{v}}\_1 & {\color{red}\mathbf{v}}\_2 & \cdots & {\color{red}\mathbf{v}}\_n \end{bmatrix}$$，則：

{% hint style="success" %}
$${\color{blue}\mathbf{u}} {\color{red}\mathbf{v}} =  \begin{bmatrix} {\color{blue}\mathbf{u}}\_1 \ {\color{blue}\mathbf{u}}\_2 \ \vdots \ {\color{blue}\mathbf{u}}\_m \end{bmatrix}  \begin{bmatrix} {\color{red}\mathbf{v}}\_1 & {\color{red}\mathbf{v}}\_2 & \cdots & {\color{red}\mathbf{v}}\_n \end{bmatrix}  =  \begin{bmatrix} {\color{blue}\mathbf{u}}\_1 {\color{red}\mathbf{v}}\_1 &  {\color{blue}\mathbf{u}}\_1 {\color{red}\mathbf{v}}\_2 & \cdots & {\color{blue}\mathbf{u}}\_1 {\color{red}\mathbf{v}}\_n \  {\color{blue}\mathbf{u}}\_2 {\color{red}\mathbf{v}}\_1 & {\color{blue}\mathbf{u}}\_2 {\color{red}\mathbf{v}}\_2 & \cdots & {\color{blue}\mathbf{u}}\_2 {\color{red}\mathbf{v}}\_n \  \vdots \  {\color{blue}\mathbf{u}}\_m {\color{red}\mathbf{v}}\_1 & {\color{blue}\mathbf{u}}\_m {\color{red}\mathbf{v}}\_2 & \cdots & {\color{blue}\mathbf{u}}\_m {\color{red}\mathbf{v}}\_n \end{bmatrix}$$
{% endhint %}

:point\_right: 比較：[「分組式」乘法](https://lochiwei.gitbook.io/math/linear/matrix/op/mult/by-groups)

* 有趣的是，這跟把 $${\color{blue}\mathbf{u}}\_1 \ \cdots \ {\color{blue}\mathbf{u}}\_m$$, $${\color{red}\mathbf{v}}\_1 \ \cdots \ {\color{red}\mathbf{v}}\_n$$ 看成一般數字然後做[矩陣乘法](https://lochiwei.gitbook.io/math/linear/matrix/op/mult)沒有差別:exclamation:


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