# 正交矩陣

[線性代數](/math/linear.md) ⟩ [向量空間](/math/linear/space.md) ⟩ [基底](/math/linear/space/basis.md) ⟩ [正交基底](/math/linear/space/basis/ortho.md) ⟩ 正交矩陣 ("<mark style="color:purple;">**orthogonal matrix**</mark>")

{% hint style="success" %}
如果一組[**基底**](/math/linear/space/basis.md)向量 ① <mark style="color:yellow;">**兩兩**</mark>[**相互垂直**](/math/linear/vec/perp.md) ② <mark style="color:yellow;">**均為**</mark>[**單位向量**](/math/linear/vec/unit.md)，若：&#x20;

* $$\mathbf{M}$$ 代表由這些基底<mark style="color:yellow;">**(列)向量**</mark>所形成的矩陣，
* 其[轉置矩陣](/math/linear/matrix/op/transpose.md) $$\mathbf{M}^T$$ 就是由這些基底<mark style="color:yellow;">**(行)向量**</mark>所形成的矩陣，

這時：&#x20;

* $$\mathbf{M}^T \mathbf{M} = \mathbf{I}$$ ，也就是 $$\mathbf{M}^T = \mathbf{M}^{-1}$$

我們稱 $$\mathbf{M}$$ 為「<mark style="color:purple;">**正交矩陣**</mark>」。
{% endhint %}

{% tabs %}
{% tab title="⭐️ 重點" %}
{% hint style="warning" %}

1. 「[正交基底](/math/linear/space/basis/ortho.md)」的向量<mark style="color:yellow;">**不須**</mark>為[單位向量](/math/linear/vec/unit.md)，但<mark style="color:purple;">**正交矩陣**</mark>的（列或行）向量<mark style="color:red;">**必須**</mark>是:exclamation:
   {% endhint %}
   {% endtab %}

{% tab title="👥 相關" %}

* [反方陣](/math/linear/matrix/op/mult/inverse.md)
* [變換法向量](/math/linear/space/transform/normal-vec.md)
  {% endtab %}

{% tab title="📗 參考" %}

* [ ] wiki ⟩  [正交矩陣](https://zh.wikipedia.org/zh-hant/正交矩阵)
  {% endtab %}
  {% endtabs %}


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