Khó-ge̍k hâng-lia̍t

Tùi chi̍t ê ${\displaystyle n}$-kai sù-hong hâng-lia̍t ${\displaystyle A}$ lâi kóng, nā-chún ū chi̍t ê ${\displaystyle n}$-kai sù-hong hâng-lia̍t ${\displaystyle B}$ sú-tek

${\displaystyle AB=BA=I_{n}$,

lán tiō kóng ${\displaystyle A}$ sī khó-ge̍k hâng-lia̍t (Hàn-jī: 可逆行列, Eng-gí: invertible matrix), kî-tiong ${\displaystyle I_{n}$ sī ${\displaystyle n}$-kai tan-ūi hâng-lia̍t, sêng-hoat sī it-poaⁿ ê hâng-lia̍t sêng-hoat; jî-chhiáⁿ chit ê ${\displaystyle B}$ sī î-it iû ${\displaystyle A}$ koat-tēng ·ê, lán tiō kā kiò-chò ${\displaystyle A}$ ê ge̍k hâng-lia̍t, kì-hō sī ${\displaystyle A^{-1}$.^[1]

Lán kóng, chi̍t ê hâng-lia̍t sī te̍k-ì hâng-lia̍t (Hàn-jī: 特異行列, Eng-gí: singular matrix) he̍k-chiá sī tòe-hòa hâng-lia̍t (Hàn-jī: 退化行列, Eng-gí: degenerate matrix) nā kiam taⁿ nā i ê hâng-lia̍t-sek téng-î lêng. Pí-lūn-kóng,

${\displaystyle A={\begin{pmatrix}3&4&7\\-1&-2&0\\1&0&7\end{pmatrix}$

sī chi̍t ê te̍k-ì hâng-lia̍t, in-ūi ${\displaystyle \det {A}=0}$.

Lán ē-sái chèng-bêng kóng, chi̍t ê hâng-lia̍t sī hui-te̍k-ì hâng-lia̍t (Hàn-jī: 非特異行列, Eng-gí: nonsingular matrix) he̍k-chiá sī tòe-hòa hâng-lia̍t (Hàn-jī: 非退化行列, Eng-gí: nondegenerate matrix) nā kiam taⁿ nā i sī khó-ge̍k hâng-lia̍t. Chhiáⁿ khòaⁿ ē-bīn ê khó-ge̍k hâng-lia̍t tēng-lí.

Lēng ${\displaystyle A}$ sī chi̍t ê tī thé K téng ê ${\displaystyle n}$-kai sù-hong hâng-lia̍t, í-hā sū-su̍t téng-kè:^[2]

1. ${\displaystyle A}$ sī khó-ge̍k hâng-lia̍t.
2. ${\displaystyle A}$ ū chi̍t ê tò-chhiú ge̍k (i.e. chū-chāi ${\displaystyle B}$ sú-tek ${\displaystyle BA=I_{n}$), ia̍h ${\displaystyle A}$ ū chi̍t ê chiàⁿ-chhiú ge̍k (i.e. chū-chāi ${\displaystyle C}$ sú-tek ${\displaystyle AC=I_{n}$); in tī lán chia kî-si̍t sio-kâng, lóng sī ${\displaystyle A}$ î-it ê ge̍k hâng-lia̍t (i.e. ${\displaystyle B=C=A^{-1}$).
3. ${\displaystyle A}$ sī hui-te̍k-ì hâng-lia̍t.
4. ${\displaystyle A}$ sī hui-tòe-hòa hâng-lia̍t.
5. ${\displaystyle \det {A}\neq 0}$.
6. ${\displaystyle A}$ kap tan-ūi hâng-lia̍t ${\displaystyle I_{n}$ chōa-téng-kè.
7. ${\displaystyle A}$ kap tan-ūi hâng-lia̍t ${\displaystyle I_{n}$ khiā-téng-kè.
8. ${\displaystyle A}$ ū ${\displaystyle n}$ ê mn̂g-sún ūi-tì.
9. ${\displaystyle A}$ móa-kai, i.e. ${\displaystyle {\text{rank}(A)=n}$
10. Hong-têng-sek ${\displaystyle A\mathbf {x} =\mathbf {0} }$ tú-tú hó ū chi̍t ê kái, sī bô-liâu kái ${\displaystyle \mathbf {x} =\mathbf {0} }$.
11. Tùi só͘-ū ${\displaystyle \mathbf {b} \in K^{n}$, hong-têng-sek ${\displaystyle A\mathbf {x} =\mathbf {b} }$ lóng tú-tú hó ū chi̍t ê kái.
12. ${\displaystyle A}$ ê hu̍t sī bô-liâu ê, i.e. ${\displaystyle \ker(A)=\{\mathbf {0} \}$.
13. ${\displaystyle A}$ ê khiā lóng sòaⁿ-sèng to̍k-li̍p.
14. ${\displaystyle A}$ ê khiā thí-khui ${\displaystyle K^{n}$.
15. ${\displaystyle A}$ ê khiā khong-kan tiō sī ${\displaystyle K^{n}$.
16. ${\displaystyle A}$ ê khiā cho͘-sêng ${\displaystyle K^{n}$ ê chi̍t cho͘ ki-té.
17. Tùi ${\displaystyle \mathbf {x} }$ kàu ${\displaystyle A\mathbf {x} }$ ê sòaⁿ-sèng siá-siōng sī chi̍t ê tùi ${\displaystyle K^{n}$ kàu ${\displaystyle K^{n}$ ê siang-siā.
18. Sò͘-jī ${\displaystyle 0}$ m̄ sī ${\displaystyle A}$ ê kò͘-iú-ta̍t.
19. ${\displaystyle A^{T}$ mā sī khó-ge̍k hâng-lia̍t. (Só͘-í, ${\displaystyle A}$ ê chōa mā lóng sòaⁿ-sèng to̍k-li̍p, mā thí-khui ${\displaystyle K^{n}$, mā cho͘-sêng ${\displaystyle K^{n}$ ê chi̍t cho͘ ki-té.)
20. Ē-sái ēng iú-hān ê ki-pún hâng-lia̍t ê sêng-chek lâi piáu-sī ${\displaystyle A}$.

Lēng ${\displaystyle A}$ sī chi̍t ê ${\displaystyle n}$-kai khó-ge̍k hâng-lia̍t, i koh ū ē-bīn kok-chióng sèng-chit:

• ${\displaystyle \left(A^{-1}\right)^{-1}=A}$.
• ${\displaystyle \left(kA\right)^{-1}=k^{-1}A^{-1}$ tùi hui-lêng sûn-liōng ${\displaystyle k}$
• ${\displaystyle \left(A^{\mathrm {T} }\right)^{-1}=\left(A^{-1}\right)^{\mathrm {T} }$.
• ${\displaystyle \det {A^{-1}=\left(\det {A}\right)^{-1}$.

Nā ${\displaystyle A}$ sī ka-kī ê ge̍k hâng-lia̍t (i.e., ${\displaystyle A=A^{-1}$) he̍k-chiá ${\displaystyle A^{2}=I}$, lán tiō kā ${\displaystyle A}$ kiò-chò tùi-ha̍p hâng-lia̍t.

Só͘-ū ê ${\displaystyle n}$-kai khó-ge̍k hâng-lia̍t hām hâng-lia̍t sêng-hoat kap chò-hóe, ē chiâⁿ-chò chi̍t ê kûn, kiò-chò ${\displaystyle n}$-kai it-poaⁿ sòaⁿ-sèng kûn.

1. ↑ Friedberg, S., Insel, A. & Spence, L. (2018). Linear Algebra (5th Edition). Pearson. ISBN 978-0134860244.
2. ↑ Stover, C. "Invertible Matrix Theorem." MathWorld. [2022-3-15]

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