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Nvmeri complexi vt matrices

Svmmarivm

Haec lectio cviqve nvmero complexo z=a+biz = a + bi matricem attribvit transformationis plani qvae per zz mvltiplicat, et demonstrat hanc correspondentiam iniectivam esse et svmmas in svmmas, prodvcta in prodvcta transformare: matrices hvivs formae exemplar fidele nvmerorvm complexorvm intra matrices qvadratas ordinis secvndi constitvvnt. In hoc exemplari vnitas imaginaria est conversio qvadrantis, cvivs qvadratvm est opposita matricis identitatis; qvadratvm modvli est determinans, conivgatvs est transposita, inversvm est matrix inversa. Forma polaris qvamqve harvm matricvm in conversionem, qvam homothetia excipit, dissolvit, et formvlae Evleri et Moivreana redevnt vt regvla compositionis conversionvm. Lectio explicat cvr hae matrices commvtabiles sint, qvamvis matrices vniverse non sint, et primo conspectv qvaternionvm Hamiltonis conclvditvr, qvorvm prodvctvm iam non est commvtativvm.

Proposita discendi

  1. Matricem mvltiplicationis per nvmervm complexvm invenire et demonstrare correspondentiam inter nvmeros complexos et matrices svmmam et prodvctvm servare.
  2. Modvlvm, conivgatvm, inversvm et formam polarem nvmeri complexi in matrices transferre, et qvamqve translationem geometrice interpretari.
  3. Potentias, inversa et radices nvmerorvm complexorvm per matrices eorvm compvtare, et in formvlis Evleri et Moivreana compositionem conversionvm agnoscere.
  4. Explicare cvr matrices nvmerorvm complexorvm inter se commvtabiles sint, demonstrare eas esse qvae cvm conversione qvadrantis commvtantvr, et per matrices confirmare prodvctvm qvaternionvm non esse commvtativvm.

Matrix mvltiplicationis per nvmervm complexvm

In lectione cvrsvs Algebra et Geometria I de nvmeris complexis qvisqve nvmervs complexvs x+yix + yi cvm pvncto (x,y)(x, y) plani identificatvs est, et demonstratvm est mvltiplicationem per ii qvodqve pvnctvm qvadrante circa originem convertere. Qvod si ita est, mvltiplicatio per nvmervm complexvm fixvm zz est transformatio plani, qvae cviqve pvncto ww pvnctvm zwzw attribvit. In lectione de matricibvs vt transformationibvs plani demonstratvm est transformationem linearem matrice determinari, cvivs colvmnae svnt imagines vectorvm (1,0)(1, 0) et (0,1)(0, 1). Inde seqvitvr qvaestio qvae hanc lectionem dirigit: qvam matricem habet mvltiplicatio per zz, et qvid lvcramvr si qvemqve nvmervm complexvm ea matrice repraesentamvs?

PropositioMatrix mvltiplicationis per z

Sit z=a+biz = a + bi nvmervs complexvs fixvs. Transformatio plani qvae qvemqve nvmervm complexvm w=x+yiw = x + yi per zz mvltiplicat linearis est; matrix eivs colvmnas habet zz et iziz, et, si z≠0z \neq 0, qvadratvm vnitarivm in qvadratvm lateris ∣z∣\abs{z} transfert.

Prodvctvm evolvimvs et in eo matricem legimvs. Figvra adhibet z=2+iz = 2 + i.

Demonstratio

  1. zw=(a+bi)(x+yi)=ax+ayi+bxi+bi2y=(ax−by)+(bx+ay)izw = (a + bi)(x + yi) = ax + ayi + bxi + b \resaltar{i^2} y = \dato{p}{\resaltar{(ax - by) + (bx + ay) i}}

    Prodvctvm per distribvtivitatem evolvimvs et pro i2i^2 ponimvs −1-1.

  2. (ax−bybx+ay)=(a−bba)(xy)\begin{pmatrix} \resaltar{ax - by} \\ \resaltar{bx + ay} \end{pmatrix} = \dato{m}{\begin{pmatrix} \resaltar{a} & \resaltar{-b} \\ \resaltar{b} & \resaltar{a} \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}}

    Pvncta vt vectores colvmnae scribimvs. Coordinatae ipsivs zwzw svnt combinationes ipsorvm xx et yy coefficientibvs fixis: est prodvctvm matricis et vectoris ipsivs ww, et transformatio igitvr linearis est.

  3. z⋅1=a+bi,z⋅i=−b+aiz \cdot \resaltar{1} = \resaltar{a + bi}, \qquad z \cdot \resaltar{i} = \resaltar{-b + ai}

    Colvmnae matricis svnt imagines ipsorvm 11 et ii. Prima est zz; altera est iz=−b+aiiz = -b + ai, ipsvm zz qvadrante conversvm.

  4. ∣iz∣=∣i∣∣z∣=∣z∣,z⋅(1+i)=z+iz\abs{iz} = \abs{i} \abs{z} = \resaltar{\abs{z}}, \qquad \resaltar{z \cdot (1 + i)} = \resaltar{z + iz}

    Qvadratvm vnitarivm, verticibvs 00, 11, 1+i1 + i et ii, in qvadrilatervm verticibvs 00, zz, z+izz + iz et iziz transformatvr. Latera eivs ab origine exevntia eandem longitvdinem ∣z∣\abs{z} habent et perpendicvlaria svnt: imago est qvadratvm lateris ∣z∣\abs{z}, respectv qvadrati vnitarii conversvm.

Matrix inventa formam habet admodvm singvlarem: dvo elementa diagonalis aeqvalia svnt, et reliqva dvo opposita. Matrix nvmeri complexi z=a+biz = a + bi, vbi aa et bb reales svnt, appellatvr matrix mvltiplicationis per zz:

M(z)=(a−bba)M(z) = \begin{pmatrix} a & -b \\ b & a \end{pmatrix}

Prima colvmna ipsivs M(z)M(z) est ipsvm zz, altera est iziz. Pars realis et pars imaginaria nvmeri complexi si separantvr, matrix in combinationem dvarvm matricvm fixarvm dissolvitvr: matricis identitatis II, qvae est matrix nvmeri 11, et matricis JJ nvmeri ii.

M(a+bi)=a(1001)+b(0−110)=aI+bJM(a + bi) = a \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + b \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} = aI + bJ

Praesertim nvmeri reales mvltiplis matricis identitatis respondent, M(a)=aIM(a) = aI, qvae svnt homothetiae centro in origine, et nvmeri imaginarii pvri bibi mvltiplis ipsivs JJ. Prima elementa tabvlae correspondentiarvm haec svnt.

M(1)=I,M(i)=J,M(−1)=−I,M(−i)=−J,M(0)=(0000)M(1) = I, \qquad M(i) = J, \qquad M(-1) = -I, \qquad M(-i) = -J, \qquad M(0) = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}