Garret Sobczyk s 2x2 Matrix Derivation

Transcription

1 Garret Sobczyk s x Matrix Derivation Krt Nalty May, 05 Abstract Using matrices to represent geometric algebras is known, bt not necessarily the best practice. While I have sed small compter programs to scan throgh candidate matrices to find representations, I have not been able to derive sch matrices from first principles. Garret Sobczyk at however, has. I repeat his derivation here, for x matrix representation of the D geometric algebra, inclding more intermediate steps for easier nderstanding. D Eclidean Geometric Algebra Basis Start with a two dimensional, Eclidean geometric algebra. We have two vector directions, and e which correspond to or standard x and y directions. Or geometric mltivector elements are scalars, vectors and e, and a single bivector e =. In this algebra, scalar mltiplication is commtative and associative, vectors sqare to scalar one, and the prodct of two vectors reslting in a bivector is anti-commtative, associative and sqares to negative one. The order sensitive mltiplication table for this algebra is shown below. e e e e e e e e e e e e -

2 Given this notation, the generic D mltivector can be written as g = α + x + ye + β e Sobczyk now introdces a linear combination of two of these basis, where he rotates the scalar and e elements 5 degrees, and applies a small scaling factor. = + e = e Adding these two basis recovers th, sbtracting these two basis recovers e. Each of the basis sqare to themselves, making these combinations idempotents like 0 0 = 0 and =. More fn, the prodct of these two basis is zero, making these combinations mtal annihilators. These two properties of projection and annihilation greatly simplify power series formlas, as fond in exponentiation. + = + e = + e = + e = e = + e + e e + e e e = = e notice below e e = = + e + e e = e + e e = e e = + e + = e + = = 0 = = Having seen how the and basis work among themselves, let s write ot the preprodct and postprodct expressions among the,, and e terms. = + e = + e = + e = + e = e = e = e = = e =

3 Now for e, e = + e e = e + e e e = e + e = e + e e e = e e = e e e e = e e Finally for the bivector e, = e e e = e + = e + = e = e = = = = e = + e e = e + e e e = e + e = e + e e e = e e = e e e e = e e So, a little smmary sheet... = e e e = e = e + = e + = e = = = = = + e / = e / + = = e = = = 0 = 0 = = = = e = e = e = e = e = e = e = e = Sobczyk now presents two matrix expressions sing geometric algebra elements. + + e e + = + e = Sobcyzk calls the right hand matrix the spectral basis for D. The next matrix expression is + = + = + = + = 3

4 Now the fn begins. Start with the tre statement for the generic mltivector g = g. [ ] [ ] g = g = + g + Being an associative algebra, we can refactor or mltiplications, keeping the order intact. g = g = [ + g ] g = g = + g ge g g e We now want to examine the middle three prodct terms. Begin by looking at each term in the center matrix. g = α + x + ye + β e g = α + x + ye + β e = x + α + βe + y e g = α + x + ye + β e = x + α βe y e g = x + α βe y e = α + x ye β e We now postmltiply by, and separate prodct terms to isolate leading and terms. Start with g. g = α + x + ye + β e = α + x + ye + β e = α + x + y + β = α + x + y + β = α + y + x + β Now, premltiply by, project and annihilate. g = α + y + x + β = α + y + x + β = α + y + 0 x + β = α + y

5 In a similar fashion, we process g g = x + α + βe + y e = x + α + βe + y e = x + α + β + y = x + β + α + y = x + β + α + y Now, premltiply by, project and annihilate. g = x + β + α + y = x + β We now rapidly do the remaining terms. g = x + α βe y e = x + α βe y e = x + α β y = x β + α y Now, premltiply by, project and annihilate. Finish with g = x β + α y = x β g = α + x ye β e = α + x ye β e = α + x y β = α + x y β = α y + x β Now, premltiply by, project and annihilate. g = α y + x β = α y 5

6 We ths have the nice reslt that g ge g g = + α + y x β x + β α y Plling ot the common factor, we have g ge α + y g ge + = x + β x β α y This real matrix is the D Eclidean geometric algebra, with respect to the spectral basis. I point ot that being an array of scalars real nmbers, not geometrical objects, this matrix commtes with the geometric nmber. α + y x β α + y x β α + y x β = x + β α y x + β α y + = x + β α y + Sobczyk ses the notation [g] for this real matrix. α + y x β [g] = x + β α y So, a little recap is in order before we proceed. g = g = + [ g ] which becomes g = + [g] Sobczyk now does a preprodct and postprodct on g. g = + [g] g = e We now sandwich mltiply by, and find e [g] 6

7 g + = e e [g] On the right hand side of the eqation, I bring the left into the matrix. g + + = + [g] + + Now I drag the across the terms, creating elements. g + e + = [g] + We now bring the term left of the [g] in the matrix from the right. g + + = + [g] + We project and annihilate. g = 0 e [g] Refactor ot the term g + = 0 0 e [g] And now absorb the nit matrix. g + = [g] e Since the [g] matrix consists of real nmbers, the mltivector and scalar array [g] commte, nlike the mltivector and the mltivector g. So, we move to the right hand side of [g], and repeat the previos steps. g + = [g] e 7

8 Bring the left into the matrix. g + + = [g] + Drag across. g + e + = [g] Bring in the right. g + + = [g] + Project and annihilate. g = [g] 0 Factor and absorb. g + = [g] We have the very pretty reslt 0 0 = [g] = [g] g + = [g] = [g] To get a stand alone formla for [g], Sobczyk develops a formla for [g], then adds the terms [g] + [g] = + [g] = [g] = [g]. To develop the formla for [g], Sobczyk once again does a sandwich prodct with conjgation with respect to. g + = [g] Doing the right hand side first, we drag across. [g] = [g] 8

9 Since [g] is a scalar array, [g] and commte. [g] = [g] = [g] Now, since sqares to one, we have [g] = [g] = [g] = [g] We now work on the left side. g + = [g] = [g] Drag across the terms. g e = [g] Bring into the colmn and row matrices. e g e = [g] Clean p the sqares e g = [g] Do the order sensitive matrix prodcts e ge g g g = [g] We now have the generic reslt g ge e ge [g] = g ge + + g g g 9

10 Consistency Check We previosly evalated g ge g g α + y = x + β x β α y We want to verify e ge g g g α + y = x + β x β α y Repeating from previosly g = α + x + ye + β e g = α + x + ye + β e = x + α + βe + y e g = α + x + ye + β e = x + α βe y e g = x + α βe y e = α + x ye β e We start with the g term. g = α + x ye β e g = α + x ye β e = α + x + y + β = α + x + y + β = α + y + x + β Project and annihilate, and get the expected reslt. Contine with the g term. g = α + y g = x + α + βe + y e g = x + α + βe + y e g = x + α β y g = x + α β y g = x β + α y 0

11 Project and annihilate, and get the expected reslt. g = x β Contine with the g term. g = x + α βe y e g = x + α βe y e g = x + α + β + y g = x + α + β + y g = x + β + α + y Project and annihilate, and get the expected reslt. Finish with the g term. g = x + β g = α + x + ye + β e g = α + x + ye + β e g = α + x y β g = α + x y β g = α y + x β Project and annihilate, and get the expected reslt. g = α y Conclsion Garret Sobczyk has demonstrated why a matrix representation for D Eclidean geometric algebra is α + y x β [g] = x + β α y