Fnte Vector Space epresentatons oss Bannster Data Asslaton esearch Centre, eadng, UK ast updated: 2nd August 2003 Contents What s a lnear vector space?......... 1 About ths docuent............ 2 1. Orthogonal systes............. 3 2. Skew systes............... 8 3. Operators spannng dfferent spaces...... 14 Index.................. 21
(c) oss Bannster, 2003 oss Bannster Data Asslaton esearch Centre Unversty of eadng, eadng, G6 6BB ross@et.rdg.ac.uk - 0 -
What s a lnear vector space? The lnear vector space representaton of felds and operators s a powerful coponent n the toolbox of the atheatcan, physcst, engneer and statstcan. It enters n the analyss of a huge range of probles. Cong under alternatve headngs of "generalsed fourer ethods", "atrx ethods", or "lnear algebra", they are especally useful n the odern day as they allow probles to be forulated n a anner that s drectly sutable for nuercal soluton by coputer. Vectors and atrces are structures that allow nforaton to be represented n a systeatc way. For our purposes a vector s a quantty that s a collecton of nuercal nforaton. Along wth the vector s ts representaton (the eanngs attached to each eleent), whch defnes a lnear vector space. Exaples arse n basc physcs, soe quanttes beng fundaentally vector n nature, such as velocty or force that exst n two- or three-densonal space. The representaton n these exaples s often of three cartesan drectons of space (the bases), and the vector can be plotted, havng a drecton and a length. In such an exaple, the partcular representaton s not rgd. The representaton can be changed by choosng dfferent cartesan drectons, by a rotaton of the orgnal axes. In the new representaton, the coponents of the vector wll have changed, not because the vector tself has changed, but because the representaton has, and replottng the vector wll yeld the sae absolute drecton and length and t dd before. Vectors can have wder, but abstract, applcatons too. Coonly a feld s represented as a vector. An exaple ght be a two-densonal feld of soe quantty constructed of a nuber of values on a fnte grd coverng the doan of the feld. A possble representaton of a vector ght then be a sequence of kronecker delta functons at each poston on the grd (these are a sple choce of bases). By assocatng each delta functon wth an eleent of the vector, the eleents becoe the values of the feld at each grd pont. As n the case of the velocty or force, the vector wll have a drecton and length, but not n two- or three- densonal space, but n an abstract N-densonal space, where N s the nuber of grd ponts n the representaton. Ths has a powerful conceptual value as all of the algebra that deals wth vectors can now be appled to felds. In ths abstract applcaton, there s the analogue of rotaton of the bass. A new 'rotated' set of axes could be constructed by proectng the feld onto an alternatve bass set. A falar exaple s the fourer decoposton of the feld, whch would yeld a set of fourer coeffcents that correspond to a representaton of plane waves, each wth a dfferent wavenuber, nstead of delta functons (the large densonalty of the vector space akes t dffcult to actually agne the fourer transforaton as a rotaton!). Thus, by perforng a fourer transfor, we have represented the sae feld (.e. vector) as an alternatve set of coeffcents. We can have ether the vector n ts real-space delta functon representaton, or the vector n ts fourer-space representaton. Just as vectors can be used to represent felds, atrces are (lnear) operators that act - 1 -
on these felds. For the purposes of ths docuent, operators ay be thought to coe n two flavours. Frst there are the transforaton or rotaton operators - these are the knds of operators that rotate the bass, transforng the representaton. Actng wth such a atrx does not fundaentally yeld a new vector, but yelds the sae vector n a new representaton, specfed by the atrx. The fourer transfor s an exaple of such an operator. The second knds of operators ght be referred to as physcal or statstcal operators, whch perfor specfc physcal or statstcal roles. Many lnear atheatcal operatons on the feld have a atrx analogue, e.g. the laplacan operator, or the convoluton operator, to nae but two. About ths docuent These notes are a descrpton of three types of representaton of vectors and physcal operators. The splest (secton 1) s applcable to systes that are represented exclusvely n ters of orthogonal bass vectors, and where both the nput and output vectors of physcal operators share the sae vector space. A generalzaton of ths (secton 2) s for systes whose vector representatons nvolve skew (non-orthogonal) bass vectors, whch requre a dfferent treatent. Fnally (secton 3), consders the treatent of operators whose nputs and outputs do not share the sae vector space. Here the technque of sngular value decoposton s ntroduced. To be clear and unabguous, dervatons are ade usng expanded (non-atrx) notaton, but alongsde the key equatons, hghlghted n boxes and usng a separate equaton nuberng syste, results are expressed n atrx notaton, whch s ore copact. In so dong, any atrces - partcularly transforaton atrces - can be nterpreted as an assebly of vectors. Ths ntroduces potental abgutes as there are alternatve conventons for dong ths. Here, when asseblng vectors nto a atrx, the conventon s used that each of the coluns of the atrx holds each of the vectors (a lesser used conventon, whch s not used here, s to use the rows). To be as general as possble, a dagger sybol s used to represent the adont of a vector, atrx or operator. For vectors and atrces, ths s the cobned acton of a T transpose and coplex conugate. If all vectors and atrces contan purely real eleents then the adont operaton s equvalent to the transpose. Applyng the adont operaton to a scalar, eans ust take the coplex conugate. Falarty wth certan atrx operatons and concepts s assued, e.g. takng the adont of a product of atrxes s equvalent to takng the adont of each ndvdually, and reversng ther order. Falarty wth the dea of an nverse atrx s also assued. - 2 -
1. epresentaton of vectors and operators n orthogonal bases In ths secton we consder the representaton of state vectors and operators n bases that have utually orthogonal coponents. The egenvectors of hertan operators are utually orthogonal and so ths secton has drect relevance to applcaton around such operators. 1.1 epresentaton of a vector In a bass coprsed of vectors, a vector v ay be expressed, 1 1 1 2 1 3 whle n an alternatve bass of 1 4 1 5 1 6 v v v v the analogues are, v v Defne a relatonshp and an nverse relatonshp between the above two sets of bases, 1 7 1 8 1 where, for the atrx equvalents, T U. Use Eq. (1.2) to fnd atrx eleents of T n Eq. (1.7), 1 9 T and slarly use Eq. (1.5) n Eq. (1.8) to fnd the atrx eleents of U, 1 10 U The rght-hand-sdes of Eqs. (1.9) and (1.10) are the adonts of each other, plyng, 1 11 1 12 v v T U U T T U.e. that the nverse transfor operator between the two orthonoral bases s the sae as the adont operator. 1 A T U 1 U 1-3 -
1.2 Change n the representaton of a vector The U and T operators have been used to transfor the bass eleents, but they also can be ade to transfor the vectors theselves. Equaton (1.1) s v n the - representaton. In order to transfor t to the -representaton, substtute Eq. (1.8) nto Eq. (1.1), 1 13 v v U U v By coparng Eq. (1.13) wth Eq. (1.4), the bracketed ter s seen to be the representaton of v n the -representaton,.e. that, 1 14 v U v T v 1 B v T v In Eq. (1.14), the property of the transfors, Eq. (1.11), has been used. In the sae way, by substtutng Eq. (1.7) nto Eq. (1.4) and coparng to Eq. (1.1), one fnds that, 1 15 v T v U v 1 C v U v In Eq. (1.15), the property, Eq. (1.11), has agan been used. 1.3 epresentaton of an operator The operators U and T that transfor between representatons are shown n secton 1.2 to be natural atrx quanttes. We now consder the converson of physcal or statstcal operators to atrx for. Here, we consder only those operators that act wthn the sae representaton, and are descrbed exclusvely by orthogonal bass ebers. et operator O act on v 2 to gve v 1. It need not be a atrx operator at ths stage - all we need know s how to operate wth O on any bass eber. 1 16 1 17 v 2 v 2 Ov 1 v 1 O In Eq. (1.17), the vectors have been expanded n the Eq. (1.2) for the orthonoralty of the vectors, -representaton. Now use - 4 -
1 18 v 2 O v 1 Ths result allows us to derve the operator O n atrx for under the - representaton. Equaton (1.18) s the expanded for of a atrx equaton, where the atrx eleents are found n the bracketed ter, 1 19 1 D v 2 O O 1 O v Thus, actng on a feld wth the operator O, s equvalent to actng on the feld expressed n the -representaton wth the atrx of the above eleents. Slarly, a atrx representaton exsts for the -representaton, n O 1 20 1 E On v 2 O v 1 The atrx eleents of Eq. (1.20) would be appled to felds expressed n the - representaton. In a slar way to dfferent possble representatons of a vector, Eqs. (1.19) and (1.20) represent the sae operator, but n two dfferent representatons. 1.4 Change n the representaton of an operator There exsts a sple relatonshp between the two representatons of the operator O, gven as Eqs. (1.19) and (1.20) n secton 1.3. Gven the atrx n one representaton, and a bass transforaton to another (as used n secton 1.1), one can fnd the atrx n the other representaton. Suppose that the atrx eleents of O n the -representaton - Eq. (1.19) - are known, what are the atrx eleents n the -representaton - Eq. (1.20) - wthout calculatng the fro scratch? Substtute Eq. (1.7) nto Eq. (1.20), 1 21 1 F On T n O O T O T T T n O T T no T The boxed expresson s the atrx equaton equvalent to Eq. (1.21) where the coluns of the atrx T specfy the -bass vectors n ters of the -bass vectors - see Eq. (1.7). Slarly, the operator n the -representaton can be found fro t n the -representaton (substtute Eq. (1.8) nto Eq. (1.19)), - 5 -
1 22 1 G O n U nonu O U O U U The coluns of the atrx specfy the -bass vectors n ters of the -bass vectors - see Eq. (1.8). 1.5 The egenrepresentaton If the vectors are egenvectors of the operator O then the -representaton of O has the specal property of beng dagonal, as shown here. If s an egenvector of O wth egenvalue then, 1 23 O The atrx eleents of Eq. (1.20) are then, On On n O n n 1 24 eanng that O s dagonal. Thus the atrx for of an operator n ts egenvector representaton s dagonal, and the dagonal eleents are ts egenvalues. et O (the operator n the -representaton as n Eq. (1.19)) be the operator n a nonegenrepresentaton (the atrx O s non-dagonal). It can be dagonalzed by transforng t - usng Eq. (1.21) - nto the bass of the egenvectors. Thus, by choosng the coluns of the atrx T to be the egenvectors of O, Eq. (1.21) would yeld the atrx eleents n the egenrepresentaton. Only the dagonal eleents n need be coputed, whch are the egenvalues, and all other eleents are zero. T If the coluns of the atrx are the representaton of the -vectors (n the - representaton) then the followng s the egenvalue equaton, n atrx for, for all egenvectors, 1 H O T TO and can be found fro a sple rearrangeent of the atrx Eq. (1.F) usng the atrx relatons of Eq. (1.A). In the above atrx equaton, the dagonal eleents of are the egenvalues. O The content of ths secton on orthonoral systes s applcable f the operator, O, s hertan. Ths s due to an portant property of hertan operators - that ther egenfunctons are orthogonal, akng the nto a sutable bass set. In the reander of secton 1.5, we prove ths property. We now defne a hertan operator. A general property of operators (n fact the defnton of the adont operator) s the followng, 1 25 O - 6 - O
1 26 O A hertan operator has the property that, 1 27 O O eanng that for such operators (fro Eq. (1.26)), 1 28 O O There are two portant propertes that can be deduced fro Eq. (1.28) - that the egenvalues of hertan operators are real, and that ther egenvectors are utually orthogonal. To show ths, consder two egenvectors of O, 1 29 O 1 30 O n n n For each of Eqs. (1.29) and (1.30), perfor an nner product wth the 'other' egenvector and do not assue yet that the egenvectors are orthogonal, n O 1 31 n 1 32 O n n n Take the adont of Eq. (1.31) and ake use of the Hertan property of Eq. (1.27), 1 33 O n n The left-hand-sdes of Eqs. (1.32) and (1.33) are dentcal. The dfference between these equatons yelds, 0 1 34 n n The sultaneous propertes of real egenvalues and orthogonal egenvectors n n are consstent wth Eq. (1.34). To satsfy ths equaton, ether the dfference of egenvalues s zero, or the nner product s zero. The dfference of egenvalues s zero when n, allowng the nner product to be nonzero. When n, the dfference between egenvalues s non-zero, whch eans that the nner product ust be zero. Another fnal pont regards the atrx eleents of hertan operators. Cobnng the hertan property of O, Eq. (1.27) wth Eq. (1.26) yelds, O O 1 35 O O Ths eans that opposte atrx eleents of hertan operators are coplex conugates of each other. If the atrx eleents are real, the the atrx s syetrc. - 7 -
2. epresentaton of vectors and operators n skew bases The atrx representaton of hertan operators (Eq. (1.27)) has specal syetry propertes. More general atrces requre a ore general treatent and often requre bass vectors that are skew to each other whch do not satsfy the orthogonalty property of Eqs. (1.2) or (1.5). In ths secton we dscuss the vector spaces that are sutable as a representaton of non-hertan atrces, but stll act wthn the sae vector space. 2.1 epresentaton of a vector Just as n secton 1, a vector can be represented n a chosen bass, v 2 1a v but, unlke n secton 1, the bass are now not assued to be utually orthogonal. Because of ths, t s useful to defne another set of bass, (a dual bass) whch goes alongsde the. The dual representaton of v s then, v 2 1b v the (represented n upper case) beng the conugate or adont of (represented n lower case). The bass set and ts dual are ntately related havng the followng b-orthogonalty relatons to ake up for the fact that the bases are skew, 2 2a 2 2b (n tensor calculus, the ters covarant and contravarant are used to dstngush between the bases). In secton 1, n effect, these two sets also exst, but they are the sae set. The b-orthogonalty property allows extracton of the expanson coeffcents n Eqs. (2.1a) and (2.1b), 2 3a 2 3b v v Analogous equatons exst for another bass set, and ts dual, v 2 4a v 2 4b 2 5a 2 5b 2 6a 2 6b v v v v v v v v - 8 -
v Defne relatonshps and nverse relatonshps between the above two sets of bases and ther duals, 2 7a T 2 7b 2 8a 2 8b T U where, for the atrx equvalents, and. Use Eqs. (2.2a), (2.2b), (2.5a) and (2.5b) to fnd atrx eleents of T, T, U and, 2 9a 2 9b 2 10a 2 10b U 1 T U T T U U T U 1 Soe of the rght hand sdes of the above four equatons are the adonts of others. Naely, Eq. (2.9a) s the conugate of Eq. (2.10b) and Eq. (2.9b) of Eq. (2.10a). Ths eans that the followng relatonshps hold, 2 11a 2 12a 2 11b 2 12b 2 Aa 2 Ab U U T T U 1 T T U 1 1 T U U 1 T U U These results ean that the nverse operator transforng between the bases s the sae as the adont of the operator actng between ts dual bases. 2.2 Change n the representaton of a vector The U, T, U and T operators have been used to transfor the bass eleents, but they also can be ade to transfor the vectors theselves. Equaton (2.1a) s n the -representaton. In order to transfor t to the -representaton, substtute Eq. (2.8a) nto Eq. (2.1a), 2 13 v v U - 9 - U v U
By coparng Eq. (2.13) wth Eq. (2.4a), the bracketed ter s seen to be the representaton of v n the -representaton,.e. that, 2 14 2 B v v T v U v T v In Eq. (2.14), the property of the transfors, Eq. (2.11b), has been used. Although we are transforng a vector fro the - to the -representaton, the T - Eq. (2.7b) - (rather than the T) operator has been used, whch transfors between the dual of these representatons. Ths s due to the fact that the bases are skew. In the sae way, by substtutng Eq. (2.7a) nto Eq. (2.4a) and coparng to Eq. (2.1a), one fnds that, 2 15 2 C v v U v T v U v In Eq. (2.15), the property of the transfors, Eq. (2.11a), has been used. These are very slar results to those proven n secton 1 for transforatons between vectors represented wth orthogonal bass sets. 2.3 epresentaton of an operator The operators U, U, T and T that transfor between representatons are shown, n secton 2.2, to be natural atrx quanttes. We now consder the converson of physcal operators to atrx for. Here we consder those operators that act wthn the sae representaton but, unlke n secton 1, the bass ebers do not have to be orthogonal. et operator O act on v 2 to gve v 1. It need not be a atrx operator at ths stage - all we need to know s how to operate wth O on any bass eber. 2 16 2 17 v 2 v 2 Ov 1 In Eq. (2.17), the vectors have been expanded n the -representaton. Ths s the sae treatent as for the orthogonal systes. Now use Eq. (2.2a) for the borthonoralty between the and vectors, 2 18 v 2-10 - v 1 O O v 1
v v v v Ths result allows us to derve the operator O n atrx for under the - representaton. Equaton (2.18) s the expanded for of a atrx equaton, where the atrx eleents are found n the bracketed ter, 2 19a O O 2 O v 1 2 Da Thus, actng on a feld wth the operator O, s equvalent to actng on the feld expressed n the -representaton wth the atrx of the above eleents. There s a slght pecularty wth the results for skew systes: although the for of the rght hand sde of Eq. (2.19a), for the atrx eleents, suggests a transference of the bass fro to, the vectors before v 1 and after v 2 the acton of O, rean n the -representaton. Slarly, a atrx representaton exsts for the -representaton, 2 19b O O 2 O v 1 2 Db Slarly for the and -representatons, On no 2 20a 2 O v 1 2 Ea On O 2 20b n 2 O v 1 2 Eb 2.4 Change n the representaton of an operator There exsts sple relatonshps between the representatons of the operator O. We wll exane the relatonshps between O n the - and -bass sets, as specfed n secton 2.3 (Eqs. (2.19a) and (2.20a) respectvely). Gven the atrx n one representaton, and a bass transforaton to another (as used n secton 2.1), one can fnd the atrx n the other representaton. Suppose that the atrx eleents of O n the -representaton - Eq. (2.19a) - are known, what are the atrx eleents of O n the -representaton - Eq. (2.20a) - wthout calculatng the fro scratch? Substtute Eqs. (2.7a) and (2.7b) nto Eq. (2.20a), On T O T n T O T n - 11 -
2 21 2 F T O T O T OT n The boxed expresson s the atrx equaton equvalent to Eq. (2.21) where the coluns of the atrx T specfy the -bass vectors n ters of the -bass vectors - see Eq. (2.7a), and the coluns of the atrx T specfy the -bass vectors n ters of the -bass vectors - see Eq. (2.7b). Slarly, the operator n the - representaton can be found fro t n the -representaton (substtute Eqs. (2.8a) and (2.8b) nto Eq. (2.19a)), 2 22 2 G O n U O U O U OnU n 2.5 The egenrepresentaton If the vectors are egenvectors of the operator O (and the are b-orthogonal to these) then the -representaton of O has the specal property of beng dagonal, as shown here. If s an egenvector of O wth egenvalue then, 2 23 O The atrx eleents of Eq. (2.20a) are then, On On no n n 2 24 eanng that O s dagonal. Thus the atrx for of an operator n ts egenvector representaton s dagonal, and the dagonal eleents are ts egenvalues. et O (the operator n the -representaton as n Eq. (2.19a)) be the operator n a nonegenrepresentaton (the atrx O s non-dagonal). It can be dagonalzed by transforng t - usng Eq. (2.21) - nto the bass of the egenvectors. Thus, by choosng the coluns of the atrx T to be the egenvectors of O, and the coluns of the atrx T to be b-orthogonal to these, Eq. (2.21) would yeld the atrx eleents n the egenrepresentaton. Only the dagonal eleents n need be coputed, whch are the egenvalues, and all other eleents are zero. If the coluns of the atrx are the representaton of the -vectors (n the - representaton) then the followng s the egenvalue equaton, n atrx for, for all egenvectors, 2 H T O T TO and can be found fro a sple rearrangeent of the atrx Eq. (2.F) usng the atrx relatons of Eqs. (2.Aa) and (2.Ab). In the above atrx equaton, the dagonal eleents of are the egenvalues. O - 12 -
Untl now, nothng has been sad about the dual space bass vectors,, apart fro beng b-orthogonal to the bass vectors. If the vectors are egenvectors of O, then the set of b-orthogonal vectors,, are found to be egenvectors of the adont operator,. O Take the egenvalue equaton for (Eq. (2.23)) and perfor an nner product wth n, but unlke n Eq. (2.24), do not assue for now that n s b-orthogonal to, no 2 25 n The adont of ths equaton s, O 2 26 n n Next, allow n to be an egenvector of O, wth egenvalue n, then perfor an nner product wth, O n n n O 2 27 n n n The left-hand-sdes of Eqs. (2.26) and (2.27) are dentcal. The dfference between these two equatons yelds, 2 28 0 n n The sultaneous propertes of the two sets of egenvalues beng related by (the egenvalues are coplex conugates of each other), and borthogonal egenvectors n n are consstent wth Eq. (2.28). To satsfy ths equaton, ether the dfference of egenvalues s zero, or the nner product s zero. The dfference of egenvalues s zero when n, allowng the nner product to be non-zero. When n, the dfference between egenvalues s non-zero, whch eans that the nner product ust be zero. - 13 -
v v v v 3. epresentaton of vectors and operators n ultple bases (wth applcaton to sngular vectors) In sectons 1 and 2, the 'nput' and 'output' vector spaces of the operator O are dentcal (the dfference between the two sectons beng the orthogonalty or b-orthogonalty of the bass ebers). In ths secton we consder the representatons of state vectors and operators necessary to descrbe the acton of operators that span two dfferent spaces. That s for those operators that act on a vector n one vector space (called the rght space snce t falls on the rght hand sde of the operator), and yeld a vector n another (called the left space). The left and rght spaces are each taken to be orthogonal spaces (see below). The separate left and rght vector spaces are necessary to descrbe the acton of physcal operators consdered n ths secton. The szes of each space can be dfferent, and so we ust defne fro the start the nuber of bases n each space. et the left space have bass ebers and the rght space have. We dd not specfy the nuber of ebers n prevous sectons snce t was constant. Another portant nuber s the rank of the operator, whch s dscussed n secton 3.5. 3.1 epresentaton of left and rght vectors A vector belongng to the left vector space can be represented n a chosen left bass, 3 1a 1 v and a vector belongng to the rght vector space can be represented n a rght bass, 3 1b 1 where the two bases are separate. Note the dfferent upper lts on the suatons. The left and rght ebers are each orthogonal sets, whch allows the coeffcents and to be found, 3 2a 3 2b 3 3a 3 3b v v n v v v n v v In an alternatve left bass and an alternatve rght bass, the analogues of the above are, 3 4a 3 4b 1 1 v v - 14 -
3 5a 3 5b 3 6a n n v v v v 3 6b Defne relatonshps and nverse relatonshps between the and n bases and the and bases, 3 7a 3 7b 3 8a 3 8b n 1 1 n 1 Tn T U 1 U n n where, for the atrx equvalents, and. Use Eqs. (3.2a), (3.2b), (3.5a) and (3.5b) to fnd atrx eleents of T, T, U and U, 3 9a 3 9b 3 10a T U 1 T n T U n n n T U 1 3 10b U Soe of the rght hand sdes of the above four equatons are the adonts of others. Naely, Eq. (3.9a) s the conugate of Eq. (3.10a) and Eq. (3.9b) of Eq. (3.10b). Ths eans that the followng relatonshps hold, 3 11a 3 12a 3 11b 3 12b 3 Aa 3 Ab U n Tn Tn U 1 n U T U 1 T U 1 U T U 1 U T - 15 -
3.2 Change n the representaton of vectors The U, T, U and T operators have been used to transfor the bass eleents, but they also can be ade to transfor the vectors theselves. Equaton (3.1) s v n the -representaton. In order to transfor t nto the -representaton, substtute Eq. (3.8a) nto Eq. (3.1a), 3 13 v 1 v n 1 U n n n 1 1 U nv By coparng Eq. (3.13) wth Eq. (3.4a), the bracketed ter s seen to be the representaton of v n the -representaton,.e. that, 3 14a 3 Ba v n 1 1 v T v U nv Tn v In Eq. (3.14a), the property of the transfors, Eq. (3.11a), has been used. In the sae way, by substtutng Eq. (3.7a) nto Eq. (3.4a) and coparng to Eq. (3.1a), one fnds the nverse relatonshp, 3 15a 3 Ca v n 1 1 v U v Tnv U n v In Eq. (3.15a), the property of the transfors, Eq. (3.11a), has been used. Slar results exst for transforatons between the two bases of the rght space. By substtutng Eq. (3.8b) nto Eq. (3.1b) and coparng to Eq. (3.4b), one fnds that, 3 14b 3 Bb v 1 1 v T v U v T v In Eq. (3.14b), the property of the transfors, Eq. (3.11b), has been used. In the sae way, by substtutng Eq. (3.7b) nto Eq. (3.4b) and coparng to Eq. (3.1b), one fnds the nverse relatonshp, n - 16 -
v 3 15b v T 1 1 v U v 3 Cb v U v In Eq. (3.15b), the property of the transfors, Eq. (3.11b), has been used. These are very slar results to those proven n secton 1, for the left and rght bases separately. 3.3 epresentaton of an operator The operators U, U, T and T that transfor between representatons are shown, n secton 3.2, to be natural atrx quanttes. We now consder the converson of physcal operators to atrx for. Here we consder those atrces that straddle two dfferent vector spaces (the operators consdered n secton 1.3 are a subset of the operators used here, whch are specal n that the two spaces happen to be the sae). et the operator O act on v to gve v. It need not be a atrx operator at ths stage - all we need to know s how to operate wth O on any rght bass eber to gve a cobnaton of left bass ebers. 3 16 3 17 1 v Ov 1 v O In Eq. (3.17), the vectors have been expanded n the -representaton n the left and rght spaces. Ths s the sae treatent as for the orthogonal systes. Now use Eq. (3.2a) for the orthonoralty of the left bass vectors, 3 18 v n n O v 1 for 1 n. Ths result allows us to derve the operator O n atrx for under the -representaton. Equaton (3.18) s the expanded for of a atrx equaton, where the atrx eleents are found n the bracketed ter, 3 19a O n n O 3 Da v O v Thus, actng on a feld wth the operator O, s equvalent to actng on the feld expressed n the -representaton wth the atrx, O, of the above eleents to yeld a vector n the -representaton. Slarly, a atrx representaton exsts for the and rght bases, -representaton n each of the left - 17 -
3 19b 3 Db O n v O v n O Expressons for the atrx eleents when one vector space s n the - representaton and the other n the -representaton are, 3 20a 3 20b 3 Ea 3 Eb O n n O v O v O n n O v O v 3.4 Change n the representaton of an operator There exsts sple relatonshps between the above representatons of the operator O. We wll exane the relatonshps between O n the - and -bass sets (n the left and rght cases), as specfed n secton 3.3. Gven the atrx n one representaton, and a bass transforaton to another (as used n secton 3.1), one can fnd the atrx n the other representaton. Suppose that the atrx eleents of O n the -representaton - Eq. (3.19a) - are known, what are the atrx eleents of O n the -representaton - Eq. (3.19b) - wthout calculatng the fro scratch? Substtute Eqs. (3.7a) and (3.7b) nto Eq. (3.19b), 3 21 3 F O n T n O 1 1 1 1 1 O T O T 1 T Tn O T Tn O T The boxed expresson s the atrx equaton equvalent to Eq. (3.21) where the coluns of the atrx T specfy the -bass vectors n ters of the -bass vectors - see Eq. (3.7a) and the coluns of the atrx T specfy the -bass vectors n ters of the -bass vectors - see Eq. (3.7b). Slarly, the operator n the - representaton can be found fro t n the -representaton (substtute Eqs. (3.8a) and (3.8b) nto Eq. (3.19a)), 3 22 O n 1 1 U n O U - 18 -
3 G O U O U The coluns of the atrx U specfy the -bass vectors n ters of the -bass vectors - see Eq. (3.8a) and the coluns of the atrx U specfy the -bass vectors n ters of the -bass vectors - see Eq. (3.8b). 3.5 The 'egenrepresentaton' (sngular vectors and sngular values) Unlke for operators that preserve the vector space, t s not possble here to wrte an egenvalue equaton (lke Eq. (1.23)). We can, however, wrte the followng, O 3 23 The vectors and take the role of the egenvectors consdered before. In the above equaton, they are called the rght and left sngular vectors. The scalar takes the role of the egenvalue. It s called the sngular value here. These sngular vectors and the sngular value satsfy other egenvalue equatons as dscussed later. If we choose to represent O n the bass vectors that satsfy Eq. (3.23), the atrx eleents O n of Eq. (3.19b) are then, O n n O n 3 24 n eanng that O s 'dagonal'. The dagonalty property s not strctly true, snce only a square atrx can be dagonal and O s a atrx. Equaton (3.24) actually says that the square subatrx of denson or (dependng on whether or respectvely s saller) at the top left of the full atrx s dagonal. Thus the atrx for of an operator n ts sngular vector representaton s 'dagonal', and the dagonal eleents are ts sngular values. et O (the operator n the -representaton as n Eq. (3.19a)) be the operator n a non-sngular vector representaton (the atrx O s non-dagonal). It can be 'dagonalzed' by transforng t - usng Eq. (3.21) - nto the bass of the left and rght sngular vectors. Thus, by choosng the coluns of the atrx T to be the left sngular vectors of O (to fnd these see Eq. (3.29b) below) and the coluns of the atrx T to be the rght sngular vectors of O (to fnd these see Eq. (3.29a) below), Eq. (3.21) would yeld the atrx eleents n the sngular representaton. Only the frst or (dependng on whch s saller) dagonal eleents n need be coputed, whch are the sngular values, and all other eleents are zero. If the coluns of the atrx T are the representaton of the -vectors (n the - representaton) and the coluns of the atrx T are the representaton of the - vectors (n the -representaton) then the followng s the sngular value equaton, Eq. (3.23), n atrx for, for all sngular vectors, 3 Ha O T T O - 19 -
and can be found fro a sple rearrangeent of the atrx Eq. (3.F) usng the atrx relatons of Eqs. (3.Aa) and (3.Ab). In the above atrx equaton, the dagonal eleents of O are the sngular values. Another sngular value equaton that can be wrtten s the followng, O 3 25 n n n whch has sngular value n. We shall show that ths s consstent wth Eq. (3.23) f the left and rght sngular vectors for orthogonal sets, and that the two sets of sngular values ( n Eq. (3.23) and n Eq. (3.25)) are real and dentcal. n n Perfor the nner product of n wth Eq. (3.23), and take the adont of the result and do not yet assue that the sngular vectors are orthogonal, O 3 26 n n Perfor the nner product of wth Eq. (3.25), O 3 27 n n n The left-hand-sdes of Eqs. (3.26) and (3.27) are dentcal. Ths yelds, 3 28 n n n The sultaneous propertes of real and dentcal sngular values and orthogonal sngular vectors ( n n and n n) are consstent wth Eq. (3.28). Note though that orthogonalty s guaranteed only when n where s the rank of the atrx operator O (or ndeed n any other representaton). has the general property that n and can be found by countng the non-zero sngular values. If a sngular value s zero then Eq. (3.28) s satsfed wthout the need for the sngular vectors to be orthogonal. Equaton (3.25) then has the followng atrx equvalent for all sngular vectors, whch can be found fro the atrx Eq. (3.Ha) usng the atrx relatons of Eqs. (3.Aa) and (3.Ab), 3 Hb O T T O The left and rght sngular vectors, and the square of the sngular values can be found fro solvng a couple of egenvalue equatons whch can be fored by consderng Eqs. (3.23) and (3.25) sultaneously (wth ), O O 2 3 29a OO 2 3 29b In atrx for, the above egenvalue equatons can be wrtten as, 3 Ia 3 Ib O O O O T T O O T T O O whch can be derved drectly fro Eqs. (3.Ha) and (3.Hb). - 20 -
Index A Adont 2 B Bass see representaton B-orthogonal bass usng egenvectors 13 B-orthogonalty 8 wth adont syste (skew systes) 13 C Coplex conugate egenvalues 13 hertan atrx eleents 7 see also adont Contravarant 8 Covarant 8 D Dagonalzaton ultple vector space operators 19 orthogonal systes 6 skew systes 12 Dual bass 8 E Egenrepresentaton orthogonal systes 6 skew systes 12 Egenvalues coplex conugates 13 real 7 H Hertan operators 3, 6 eft space 14 near vector space exaples 1 left and rght 14 orthogonal systes 3 skew systes 8 M Matrces 1 for of operator 5, 11, 17 Multple bases 14 O Operators change n the representaton 5, 11, 18 atrx for of 5, 11, 17 ultple vector space operators 17 orthogonal and hertan systes 4 physcal operator 2 skew systes 10 spannng two vector spaces 14 transforaton operator 1 Orthogonal bass 3 usng egenvectors 7 usng left and rght sngular vectors 20 ank 20 epresentaton exaples 1 ght space 14 otaton 1 S Sngular values 19 real 20 Sngular vectors 14, 19 orthogonal 20 Skew bases 8 T Transforatons between orthogonal bass 3 between orthogonal bass of left and rght vectors 15 between skew bases 9 rotaton 1 Transpose see adont V Vectors 1 change n the representaton of 4, 9, 16 ultple vector space operators 14 orthogonal systes 3 skew systes 8-21 -