Kiōng-tam choán-tì

Kiōng-tam choán-tì (Hàn-jī: 共擔轉置, Eng-gí: conjugate transpose) sī chiam-tùi hâng-lia̍t ê chi̍t chióng chhau-chok, kán-tan lâi kóng, i pau-hâm chi̍t pái ho̍k-kiōng-tam kap chi̍t pái choán-tì.

Chi̍t ê ho̍k hâng-lia̍t ${\displaystyle A}$ ê kiōng-tam choán-tì, kì-hō ū chiok chē chióng:

• Tī sòaⁿ-sèng tāi-sò͘ tiong, tiāⁿ-tiāⁿ ē siá-chò ${\displaystyle A^{*}$ ia̍h-sī ${\displaystyle A^{\mathrm {H} }$.^[1][2]
• Tī liōng-chú le̍k-ha̍k tiong, tiāⁿ-tiāⁿ ē siá-chò ${\displaystyle A^{\dagger }$, ēng Eng-gí tha̍k-chò “A dagger”.^[3]
• Ū sî-chūn ē siá-chò ${\displaystyle A^{+}$, sui-jiân-kóng chit ê hû-hō khah chia̍p sī piáu-sī Moore–Penrose pseudoinverse.

${\displaystyle A}$ ê kiōng-tam choán-tì lán ē-sái án-ne lâi tēng-gī:$${\displaystyle \left(A^{\mathrm {H} }\right)_{jk}={\overline {A_{kj},}$$kî-tiong iàu-sò͘ ${\displaystyle {\overline {A_{kj}$ téng-koân chi̍t hoâiⁿ ê ì-sù sī ${\displaystyle A_{kj}$ ê ho̍k-kiōng-tam.

Ká-sú-kóng lán beh kè-sǹg hâng-lia̍t

${\displaystyle A={\begin{pmatrix}1&-2-i&5\\1+i&i&4-2i\end{pmatrix}$

ê kiōng-tam choán-tì, lán tio̍h seng sǹg i ê choán-tì:

${\displaystyle A^{\mathrm {T} }={\begin{pmatrix}1&1+i\\-2-i&i\\5&4-2i\end{pmatrix},}$

koh sǹg i ê ho̍k-kiōng-tam:

${\displaystyle A^{\mathrm {H} }={\begin{pmatrix}1&1-i\\-2+i&-i\\5&4+2i\end{pmatrix}.}$

Chia-ê mài-chheng ì-sù chha-put-to lóng sio-kâng:

• kiōng-tam choán-tì (Hàn-jī: 共擔轉置, Eng-gí: conjugate transpose)^[2]
• Hermite kiōng-tam (Hàn-jī: Hermite共擔, Eng-gí: Hermitian conjugate)
• Hermite choán-tì (Hàn-jī: Hermite轉置, Eng-gí: Hermitian transpose)
• adjoint (Eng-gí: adjoint)^[2][3]

• Si̍t hâng-lia̍t ${\displaystyle A}$ ê kiōng-tam choán-tì tiō sī i ê choán-tì: ${\displaystyle A^{\mathrm {H} }=A^{\mathrm {T} }$.
• Tùi jīm-hô nn̄g ê pêⁿ-tōa ê hâng-lia̍t ${\displaystyle A}$ kap ${\displaystyle B}$, lán ū ${\displaystyle (A+B)^{\mathrm {H} }=A^{\mathrm {H} }+B^{\mathrm {H} }$ .
• Tùi jīm-hô ho̍k-cha̍p-sò͘ ${\displaystyle z}$ kap jīm-hô ${\displaystyle m\times n}$ hâng-lia̍t ${\displaystyle A}$, lán ū ${\displaystyle (zA)^{\mathrm {H} }={\overline {z}A^{\mathrm {H} }$ .
• Tùi jīm-hô ${\displaystyle m\times n}$ hâng-lia̍t ${\displaystyle A}$ kap jīm-hô ${\displaystyle n\times p}$ hâng-lia̍t ${\displaystyle B}$, lán ū ${\displaystyle \left(AB\right)^{\mathrm {H} }=B^{\mathrm {H} }A^{\mathrm {H} }$ . Ài chù-ì, sūn-sī tian-tò-péng ·ah.
• Tùi jīm-hô ${\displaystyle m\times n}$ hâng-lia̍t ${\displaystyle A}$, lán ū ${\displaystyle \left(A^{\mathrm {H} }\right)^{\mathrm {H} }=A}$, i.e. kiōng-tam choán-tì sī chi̍t chióng tùi-ha̍p.

1. ↑ “conjugate transpose”. planetmath.org. [2020-09-08].
2. ↑ ^{2.0 2.1 2.2} Friedberg, S., Insel, A. & Spence, L. (2018). Linear Algebra (5th Edition). Pearson. ISBN 978-0134860244.
3. ↑ ^{3.0 3.1} Shankar, R. (2012). Principles of Quantum Mechanics (2nd Edition). Springer.

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