Matrix Groups

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\(\def\vs{{\it vs. }} \def\cf{{\it cf. }} \def\viz{{\it viz. }} \def\ie{{\it i.e. }} \def\etc{{\it etc. }} \def\eg{{\it e.g. }} \def\etal{{\it et al .}} \def\via{{\it via }} \def\adhoc{{\it ad hoc }} \def\apriori{{\it apriori }} \def\Afrak{\mathfrak{A}} \def\Bfrak{\mathfrak{B}} \def\Cfrak{\mathfrak{C}} \def\Dfrak{\mathfrak{D}} \def\Efrak{\mathfrak{E}} \def\Ffrak{\mathfrak{F}} \def\Gfrak{\mathfrak{G}} \def\Hfrak{\mathfrak{H}} \def\Ifrak{\mathfrak{I}} \def\Jfrak{\mathfrak{J}} \def\Kfrak{\mathfrak{K}} \def\Lfrak{\mathfrak{L}} \def\Mfrak{\mathfrak{M}} \def\Nfrak{\mathfrak{N}} \def\Ofrak{\mathfrak{O}} \def\Pfrak{\mathfrak{P}} \def\Qfrak{\mathfrak{Q}} \def\Rfrak{\mathfrak{R}} \def\Sfrak{\mathfrak{S}} \def\Tfrak{\mathfrak{T}} \def\Ufrak{\mathfrak{U}} \def\Vfrak{\mathfrak{V}} \def\Wfrak{\mathfrak{W}} \def\Xfrak{\mathfrak{X}} \def\Yfrak{\mathfrak{Y}} \def\Zfrak{\mathfrak{Z}} \def\afrak{\mathfrak{a}} \def\bfrak{\mathfrak{b}} \def\cfrak{\mathfrak{c}} \def\dfrak{\mathfrak{d}} \def\efrak{\mathfrak{e}} \def\ffrak{\mathfrak{f}} \def\gfrak{\mathfrak{g}} \def\hfrak{\mathfrak{h}} \def\ifrak{\mathfrak{i}} \def\jfrak{\mathfrak{j}} \def\kfrak{\mathfrak{k}} \def\lfrak{\mathfrak{l}} \def\mfrak{\mathfrak{m}} \def\nfrak{\mathfrak{n}} \def\ofrak{\mathfrak{o}} \def\pfrak{\mathfrak{p}} \def\qfrak{\mathfrak{q}} \def\rfrak{\mathfrak{r}} \def\sfrak{\mathfrak{s}} \def\tfrak{\mathfrak{t}} \def\ufrak{\mathfrak{u}} \def\vfrak{\mathfrak{v}} \def\wfrak{\mathfrak{w}} \def\xfrak{\mathfrak{x}} \def\yfrak{\mathfrak{y}} \def\zfrak{\mathfrak{z}} \def\AA{\mathbb{A}} \def\BB{\mathbb{B}} \def\CC{\mathbb{C}} \def\DD{\mathbb{D}} \def\EE{\mathbb{E}} \def\FF{\mathbb{F}} \def\GG{\mathbb{G}} \def\HH{\mathbb{H}} \def\II{\mathbb{I}} \def\JJ{\mathbb{J}} \def\KK{\mathbb{K}} \def\LL{\mathbb{L}} \def\MM{\mathbb{M}} \def\NN{\mathbb{N}} \def\OO{\mathbb{O}} \def\PP{\mathbb{P}} \def\QQ{\mathbb{Q}} \def\RR{\mathbb{R}} \def\SS{\mathbb{S}} \def\TT{\mathbb{T}} \def\UU{\mathbb{U}} \def\VV{\mathbb{V}} \def\WW{\mathbb{W}} \def\XX{\mathbb{X}} \def\YY{\mathbb{Y}} \def\ZZ{\mathbb{Z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\Ap{A^\prime} \def\Bp{B^\prime} \def\Cp{C^\prime} \def\Dp{D^\prime} \def\Ep{E^\prime} \def\Fp{F^\prime} \def\Gp{G^\prime} \def\Hp{H^\prime} \def\Ip{I^\prime} \def\Jp{J^\prime} \def\Kp{K^\prime} \def\Lp{L^\prime} \def\Mp{M^\prime} \def\Mp{N^\prime} \def\Op{O^\prime} \def\Pp{P^\prime} \def\Qp{Q^\prime} \def\Rp{R^\prime} \def\Sp{S^\prime} \def\Tp{T^\prime} \def\Up{U^\prime} \def\Vp{V^\prime} \def\Wp{W^\prime} \def\Xp{X^\prime} \def\Yp{Y^\prime} \def\Zp{Z^\prime} \def\ap{a^\prime} \def\bp{b^\prime} \def\cp{c^\prime} \def\dprime{d^\prime} \def\ep{e^\prime} \def\fp{f^\prime} \def\gp{g^\prime} \def\hp{h^\prime} \def\ip{i^\prime} \def\jp{j^\prime} \def\kp{k^\prime} \def\lp{l^\prime} \def\mp{m^\prime} \def\np{n^\prime} \def\op{o^\prime} \def\pp{p^\prime} \def\qp{q^\prime} \def\rp{r^\prime} \def\sp{s^\prime} \def\tp{t^\prime} \def\up{u^\prime} \def\vp{v^\prime} \def\wp{w^\prime} \def\xp{x^\prime} \def\yp{y^\prime} \def\zp{z^\prime} \def\App{A^{\prime\prime}} \def\Bpp{B^{\prime\prime}} \def\Cpp{C^{\prime\prime}} \def\Dpp{D^{\prime\prime}} \def\Epp{E^{\prime\prime}} \def\Fpp{F^{\prime\prime}} \def\Gpp{G^{\prime\prime}} \def\Hpp{H^{\prime\prime}} \def\Ipp{I^{\prime\prime}} \def\Jpp{J^{\prime\prime}} \def\Kpp{K^{\prime\prime}} \def\Lpp{L^{\prime\prime}} \def\Mpp{M^{\prime\prime}} \def\Mpp{N^{\prime\prime}} \def\Opp{O^{\prime\prime}} \def\Ppp{P^{\prime\prime}} \def\Qpp{Q^{\prime\prime}} \def\Rpp{R^{\prime\prime}} \def\Spp{S^{\prime\prime}} \def\Tpp{T^{\prime\prime}} \def\Upp{U^{\prime\prime}} \def\Vpp{V^{\prime\prime}} \def\Wpp{W^{\prime\prime}} \def\Xpp{X^{\prime\prime}} \def\Ypp{Y^{\prime\prime}} \def\Zpp{Z^{\prime\prime}} \def\app{a^{\prime\prime}} \def\bpp{b^{\prime\prime}} \def\cpp{c^{\prime\prime}} \def\dpp{d^{\prime\prime}} \def\epp{e^{\prime\prime}} \def\fpp{f^{\prime\prime}} \def\gpp{g^{\prime\prime}} \def\hpp{h^{\prime\prime}} \def\ipp{i^{\prime\prime}} \def\jpp{j^{\prime\prime}} \def\kpp{k^{\prime\prime}} \def\lpp{l^{\prime\prime}} \def\mpp{m^{\prime\prime}} \def\npp{n^{\prime\prime}} \def\opp{o^{\prime\prime}} \def\ppp{p^{\prime\prime}} \def\qpp{q^{\prime\prime}} \def\rpp{r^{\prime\prime}} \def\spp{s^{\prime\prime}} \def\tpp{t^{\prime\prime}} \def\upp{u^{\prime\prime}} \def\vpp{v^{\prime\prime}} \def\wpp{w^{\prime\prime}} \def\xpp{x^{\prime\prime}} \def\ypp{y^{\prime\prime}} \def\zpp{z^{\prime\prime}} \def\abar{\overline{a}} \def\bbar{\overline{b}} \def\cbar{\overline{c}} \def\dbar{\overline{d}} \def\ebar{\overline{e}} \def\fbar{\overline{f}} \def\gbar{\overline{g}} \def\ibar{\overline{i}} \def\jbar{\overline{j}} \def\kbar{\overline{k}} \def\lbar{\overline{l}} \def\mbar{\overline{m}} \def\nbar{\overline{n}} \def\obar{\overline{o}} \def\pbar{\overline{p}} \def\qbar{\overline{q}} \def\rbar{\overline{r}} \def\sbar{\overline{s}} \def\tbar{\overline{t}} \def\ubar{\overline{u}} \def\vbar{\overline{v}} \def\wbar{\overline{w}} \def\xbar{\overline{x}} \def\ybar{\overline{y}} \def\zbar{\overline{z}} \def\Abar{\overline{A}} \def\Bbar{\overline{B}} \def\Cbar{\overline{C}} \def\Dbar{\overline{D}} \def\Ebar{\overline{E}} \def\Fbar{\overline{F}} \def\Gbar{\overline{G}} \def\Hbar{\overline{H}} \def\Ibar{\overline{I}} \def\Jbar{\overline{J}} \def\Kbar{\overline{K}} \def\Lbar{\overline{L}} \def\Mbar{\overline{M}} \def\Nbar{\overline{N}} \def\Obar{\overline{O}} \def\Pbar{\overline{P}} \def\Qbar{\overline{Q}} \def\Rbar{\overline{R}} \def\Sbar{\overline{S}} \def\Tbar{\overline{T}} \def\Ubar{\overline{U}} \def\Vbar{\overline{V}} \def\Wbar{\overline{W}} \def\Xbar{\overline{X}} \def\Ybar{\overline{Y}} \def\Zbar{\overline{Z}} \def\aunder{\underline{a}} \def\bunder{\underline{b}} \def\cunder{\underline{c}} \def\dunder{\underline{d}} \def\eunder{\underline{e}} \def\funder{\underline{f}} \def\gunder{\underline{g}} \def\hunder{\underline{h}} \def\iunder{\underline{i}} \def\junder{\underline{j}} \def\kunder{\underline{k}} \def\lunder{\underline{l}} \def\munder{\underline{m}} \def\nunder{\underline{n}} \def\ounder{\underline{o}} \def\punder{\underline{p}} \def\qunder{\underline{q}} \def\runder{\underline{r}} \def\sunder{\underline{s}} \def\tunder{\underline{t}} \def\uunder{\underline{u}} \def\vunder{\underline{v}} \def\wunder{\underline{w}} \def\xunder{\underline{x}} \def\yunder{\underline{y}} \def\zunder{\underline{z}} \def\Aunder{\underline{A}} \def\atilde{\widetilde{a}} \def\btilde{\widetilde{b}} \def\ctilde{\widetilde{c}} \def\dtilde{\widetilde{d}} \def\etilde{\widetilde{e}} \def\ftilde{\widetilde{f}} \def\gtilde{\widetilde{g}} \def\htilde{\widetilde{h}} \def\itilde{\widetilde{i}} \def\jtilde{\widetilde{j}} \def\ktilde{\widetilde{k}} \def\ltilde{\widetilde{l}} \def\mtilde{\widetilde{m}} \def\ntilde{\widetilde{n}} \def\otilde{\widetilde{o}} \def\ptilde{\widetilde{p}} \def\qtilde{\widetilde{q}} \def\rtilde{\widetilde{r}} \def\stilde{\widetilde{s}} \def\ttilde{\widetilde{t}} \def\utilde{\widetilde{u}} \def\vtilde{\widetilde{v}} \def\wtilde{\widetilde{w}} \def\xtilde{\widetilde{x}} \def\ytilde{\widetilde{y}} \def\ztilde{\widetilde{z}} \def\Atilde{\widetilde{A}} \def\Btilde{\widetilde{B}} \def\Ctilde{\widetilde{C}} \def\Dtilde{\widetilde{D}} \def\Etilde{\widetilde{E}} \def\Ftilde{\widetilde{F}} \def\Gtilde{\widetilde{G}} \def\Htilde{\widetilde{H}} \def\Itilde{\widetilde{I}} \def\Jtilde{\widetilde{J}} \def\Ktilde{\widetilde{K}} \def\Ltilde{\widetilde{L}} \def\Mtilde{\widetilde{M}} \def\Ntilde{\widetilde{N}} \def\Otilde{\widetilde{O}} \def\Ptilde{\widetilde{P}} \def\Qtilde{\widetilde{Q}} \def\Rtilde{\widetilde{R}} \def\Stilde{\widetilde{S}} \def\Ttilde{\widetilde{T}} \def\Utilde{\widetilde{U}} \def\Vtilde{\widetilde{V}} \def\Wtilde{\widetilde{W}} \def\Xtilde{\widetilde{X}} \def\Ytilde{\widetilde{Y}} \def\Ztilde{\widetilde{Z}} \def\Alphatilde{\widetilde{\Alpha}} \def\Betatilde{\widetilde{\Beta}} \def\Gammatilde{\widetilde{\Gamma}} \def\Deltatilde{\widetilde{\Delta}} \def\Epsilontilde{\widetilde{\Epsilon}} \def\Zetatilde{\widetilde{\Zeta}} \def\Etatilde{\widetilde{\Eta}} \def\Thetatilde{\widetilde{\Theta}} \def\Iotatilde{\widetilde{\Iota}} \def\Kappatilde{\widetilde{\Kappa}} \def\Lambdatilde{\widetilde{\Lamdba}} \def\Mutilde{\widetilde{\Mu}} \def\Nutilde{\widetilde{\Nu}} \def\Xitilde{\widetilde{\Xi}} \def\Omicrontilde{\widetilde{\Omicron}} \def\Pitilde{\widetilde{\Pi}} \def\Rhotilde{\widetilde{\Rho}} \def\Sigmatilde{\widetilde{\Sigma}} \def\Tautilde{\widetilde{\Tau}} \def\Upsilontilde{\widetilde{\Upsilon}} \def\Phitilde{\widetilde{\Phi}} \def\Chitilde{\widetilde{\Chi}} \def\Psitilde{\widetilde{\Psi}} \def\Omegatilde{\widetilde{\Omega}} \def\alphatilde{\widetilde{\alpha}} \def\betatilde{\widetilde{\beta}} \def\gammatilde{\widetilde{\gamma}} \def\deltatilde{\widetilde{\delta}} \def\epsilontilde{\widetilde{\epsilon}} \def\zetatilde{\widetilde{\zeta}} \def\etatilde{\widetilde{\eta}} \def\thetatilde{\widetilde{\theta}} \def\iotatilde{\widetilde{\iota}} \def\kappatilde{\widetilde{\kappa}} \def\lambdatilde{\widetilde{\lamdba}} \def\mutilde{\widetilde{\mu}} \def\nutilde{\widetilde{\nu}} \def\xitilde{\widetilde{\xi}} \def\omicrontilde{\widetilde{\omicron}} \def\pitilde{\widetilde{\pi}} \def\rhotilde{\widetilde{\rho}} \def\sigmatilde{\widetilde{\sigma}} \def\tautilde{\widetilde{\tau}} \def\upsilontilde{\widetilde{\upsilon}} \def\phitilde{\widetilde{\phi}} \def\chitilde{\widetilde{\chi}} \def\psitilde{\widetilde{\psi}} \def\omegatilde{\widetilde{\omega}} \def\Alphabar{\bar{\Alpha}} \def\Betabar{\bar{\Beta}} \def\Gammabar{\bar{\Gamma}} \def\Deltabar{\bar{\Delta}} \def\Epsilonbar{\bar{\Epsilon}} \def\Zetabar{\bar{\Zeta}} \def\Etabar{\bar{\Eta}} \def\Thetabar{\bar{\Theta}} \def\Iotabar{\bar{\Iota}} \def\Kappabar{\bar{\Kappa}} \def\Lambdabar{\bar{\Lamdba}} \def\Mubar{\bar{\Mu}} \def\Nubar{\bar{\Nu}} \def\Xibar{\bar{\Xi}} \def\Omicronbar{\bar{\Omicron}} \def\Pibar{\bar{\Pi}} \def\Rhobar{\bar{\Rho}} \def\Sigmabar{\bar{\Sigma}} \def\Taubar{\bar{\Tau}} \def\Upsilonbar{\bar{\Upsilon}} \def\Phibar{\bar{\Phi}} \def\Chibar{\bar{\Chi}} \def\Psibar{\bar{\Psi}} \def\Omegabar{\bar{\Omega}} \def\alphabar{\bar{\alpha}} \def\betabar{\bar{\beta}} \def\gammabar{\bar{\gamma}} \def\deltabar{\bar{\delta}} \def\epsilonbar{\bar{\epsilon}} \def\zetabar{\bar{\zeta}} \def\etabar{\bar{\eta}} \def\thetabar{\bar{\theta}} \def\iotabar{\bar{\iota}} \def\kappabar{\bar{\kappa}} \def\lambdabar{\bar{\lamdba}} \def\mubar{\bar{\mu}} \def\nubar{\bar{\nu}} \def\xibar{\bar{\xi}} \def\omicronbar{\bar{\omicron}} \def\pibar{\bar{\pi}} \def\rhobar{\bar{\rho}} \def\sigmabar{\bar{\sigma}} \def\taubar{\bar{\tau}} \def\upsilonbar{\bar{\upsilon}} \def\phibar{\bar{\phi}} \def\chibar{\bar{\chi}} \def\psibar{\bar{\psi}} \def\omegabar{\bar{\omega}} \def\del{\partial} \def\delbar{\overline{\partial}} \def\Cech{\check{C}} \def\half{\frac{1}{2}} \def\defeq{\mathrel{\mathop:}=} \def\alg{\mathrm{alg}} \def\Alt{\mathrm{Alt}} \def\Amp{\mathrm{Amp}} \def\Arg{\mathrm{Arg}} \def\an{\mathrm{an}} \def\anti{\mathrm{anti}} \def\Ap{\mathrm{Ap}} \def\arcsinh{\mathrm{arcsinh\hspace{0.07cm}}} \def\arccosh{\mathrm{arccosh\hspace{0.07cm}}} \def\arctanh{\mathrm{arctanh\hspace{0.07cm}}} \def\arccsch{\mathrm{arccsch\hspace{0.07cm}}} \def\arcsech{\mathrm{arcsech\hspace{0.07cm}}} \def\arccoth{\mathrm{arccoth\hspace{0.07cm}}} \def\arccsc{\mathrm{arccsc\hspace{0.07cm}}} 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\def\log{\mathrm{log}} \def\Log{\mathrm{Log}} \def\Li{\mathrm{Li}} \def\min{\mathrm{min}} \def\Mon{\mathrm{Mon}} \def\Nef{\mathrm{Nef}} \def\NS{\mathrm{NS}} \def\Oa{\mathcal{O}_A} \def\Ox{\mathcal{O}_X} \def\Oz{\mathcal{O}_Z} \def\Perp{\mathrm{Perp}} \def\Pic{\mathrm{Pic}} \def\Proj{\mathrm{Proj}} \def\rank{\mathrm{rank}} \def\Rat{\mathrm{Rat}} \def\Real{\mathrm{Re}} \def\reg{\mathrm{reg}} \def\Res{\mathrm{Res}} \def\res{\mathrm{res}} \def\Ric{\mathrm{Ric}} \def\sech{\mathrm{sech}\hspace{0.07cm}} \def\Span{\mathrm{Span}} \def\Spec{\mathrm{Spec}} \def\sing{\mathrm{sing}} \def\Singx{\mathrm{Sing}(X)} \def\sheafKer{\mathcal{\Ker}} \def\sheafIm{\mathcal{\Im}} \def\Span{\mathrm{Span}} \def\Spin{\mathrm{Spin}} \def\Str{\mathrm{Str}} \def\td{\mathrm{td}} \def\tr{\mathrm{tr}} \def\Todd{\mathrm{Todd}} \def\tor{\mathrm{tor}} \def\trdeg{\mathrm{trdeg}} \def\Zar{\mathrm{Zar}} \def\ZFLS{\mathrm{ZFLS}} \usepackage{tikz} \usepackage{tkz-graph} \usepackage{tkz-euclide} \usetikzlibrary{patterns} 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Section 9.2 Matrix Groups

Subsection 9.2.1 Symmetry, Again

A major theme of this course is symmetry. When I defined linear transformations, I defined them to be transformations that preserved something. Geometrically, they preserved linear subspaces: each linear subspace is sent to another linear subspace. Algebraically, they preserved the linear operations of addition and scalar multiplication: I can do vector addition or scalar multiplication before or after the transformation and the result is the same. This the general notion of symmetry that I want to develop in this course: transformations that preserve something (either algebraic or geometric).

This general idea of symmetry continues to develop as the course moves deeper into the discipline of linear algebra. The classification of matrices, at least geometrically, is almost always a question of symmetry. In this section, I’m going to define some new classes of matrices in terms of their symmetries: in terms of what they preserve.

Recall the notation I’ve already established for matrix sets and groups (with real coefficients).

The set of \(n \times m\) matrices is written \(M_{n,m}(\RR)\text{.}\)
The set of square (\(n \times n\)) matrices is written \(M_n(\RR)\text{.}\)
The group of invertible matrices is called the general linear group and written \(GL_n(\RR)\) or \(GL(n)\text{.}\)

Subsection 9.2.2 Transformations that Preserve Size and Orientation

Determinants measure two geometric properties: size and orientation. These are both potential symmetries. If a matrix has determinant \(\pm 1\text{,}\) I know that it preserves size. If a matrix has positive determinant, I know that it preserves orientation. If a matrix has determinant one, it preserves both.

Definition 9.2.1.

The group of matrices of determinant one with real coefficients is called the Special Linear Group and is written \(SL_n(\RR)\) or \(SL(n)\text{.}\)

Definition 9.2.2.

The group of orthogonal \(n \times n\) matrices is called the Orthogonal Group and is written \(O_n(\RR)\) or \(O(n)\text{.}\) The group of orthogonal matrices, which also preserve orientation (have determinant one) is called the Special Orthogonal Group and is written \(SO_n(\RR)\) or \(SO(n)\text{.}\)