January 2020
Beginner to intermediate
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Japanese
Content preview from データサイエンス設計マニュアル
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8.2 行列演算の可視化 229
図 8 -8 リ
ンカーンの逆行列（左）は人間のようには見えないが、MM
−1
を計算すれば、主対角線が描か
れる。ただし、精度の問題から、主対角線以外にも 0 ではない小さな値になる部分がある
成される。浮動小数点数演算は本質的に不正確であり、加算と乗算を反復的に行う逆行列の乗算のようなア
ルゴリズムでは、処理の過程で生まれた誤差が蓄積される。
逆行列はどのように求めるのだろうか。2 × 2 行列 A の逆行列は次のように求められる。
A
−
1
=
a b
c d
−1
=
1
ad − bc
d −b
−c a
より一般的に、ガウスの消去法で線形方程式を解いて逆行列を求める方法がある。
こ
の逆行列の公式をよく見ると、対角線の積が等しければ、つまり ad = bc なら、0 による除算が行われ
ることに注意しよう。そのような行列は逆行列を持たず、特異行列と呼ばれる。数値を 0 で除算できないの
と同じように、特異行列の逆行列は求められない。
逆行列がある行列は正則行列と呼ばれる。この性質を持つ行列の方が扱いやすい。行列は、行列式が 0 で
なければ非特異である。2 × 2 行列の場合、行列式は対角線上の要素の積の差であり、これはまさしく逆行
列を求める公式の分母である。
さらに、行列式が定義されるのは正方行列だけであって、行列式が 0 でない限り、また、そのときに限
り正方行列は逆行列を持つ。行列式の計算コストは O(n
3
) なので、大規模な行列では相当なコストになる。
実際、ガウスの消去法を使って行列から直接逆行列を作ろうとするのと同じコストがかかる。 ...
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