Normal Mode Decomposition of 2N 2N symplectic matrices

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1 David Rubin February 8, 008 Normal Mode Decomposition of N N symplectic matrices Normal mode decomposition of a 4X4 symplectic matrix is a standard technique for analyzing transverse coupling in a storage ring. We generalize the decomposition to any nxn symplectic matrix T and derive the transformation W from lab coordinates to normal mode coordinates U. That is T W UW ( where U is block diagonal and real and we construct the real matrix W with the form γ I C C... W C γ I C 3... C C 3 γ 3 I... ( I is the x identity, and C, C, C etc are x. (If for example, n, then γ γ and C C. The matrix U A B... can be decomposed as U Y ZY (3 where with and Z(θ, θ,..., θ n R(θ R(θ... (4 ( cos θ sin θ R(θ sin θ cos θ (5 Y G G..., (6

2 ( and G i α i βi βi. Since standard techniques exist for diagonalizing square matrices and identifying eigenvalues and eigenvectors, we begin by doing just that. T V DV, (7 where T is the n x n symplectic matrix, D is the diagonal matrix of eigenvalues, and V is the matrix constructed from the eigenvectors. Since T is symplectic, the eigenvalues and eigenvectors appear as unimodular, complex conjugate pairs, λ i, λ i and v i and v i. Then D can be written in the form d(θ ( D 0 d(θ 0... e 0 0 d(θ 3... where d(θ iθ 0 0 e iθ. (8 The n columns of the matrix V are the n eigenvectors v i. The eigenvectors are not unique, but may be multiplied by an arbitrary complex number. That is, v i ρ i e iφ i v i and v i ρ i e iφ i v i. If V 0 v v v v... v n v n, then V ( ρ, φ V 0 D(ρ, ρ,...ρ n, φ, φ,..., φ n ρ d(φ V 0 0 ρ d(φ ρ 3 d(φ 3... Note that V ( ρ, φ effects the transformation of Equation 7 for any real numbers ρ i and φ i. We transform from a complex to a real basis with K where the real matrix Z (Equation 4 is related to the complex matrix D (Equation 8 by the similarity transformation Z(θ, θ, θ 3 KD(θ, θ, θ 3 K (9 where and K k k 0 (0 0 0 k k ( i i (

3 W-matrix To construct W and U from V and D, we use Equations, 3 and 9 to write T W UW V DV V 0 D( ρ, φ ( K K D( θ ( K K D ( ρ, φv 0 ( V 0 K K D( ρ, φ ( K K D( θ ( KK D ( ρ, φ ( KK V0 ( V 0 K Z( ρ, φz( θz ( ρ, φ ( K V0 V ( ρ, φz( θv ( ρ, φ Now since the columns of V 0 are complex conjugate pairs, V 0 K is real. The Z matrices are similarly constructed to be real and therefore V is real. So far we have W UW V ZV W Y Z( θy V Z( θv V W Y where we have used Equation 3. Next we determine the parameters ρ and φ. We choose ρ so that V will be symplectic. In particular, if we write V in terms of the X matrices V j i then V V V... V V... V 0 V 0... V 0 V 0... ρ R(φ ρ R(φ ρ V 0 R(φ ρ V 0 R(φ... ρ V 0 R(φ ρ V 0 R(φ... Symplecticity constrains the sums of determinants of V i j so that i V i j 3

4 i i ρ j ρ j V 0 j i R(φ j ρ j V 0 j i n i V 0 j i In order to determine the order of the conjugate columns of V, and finally the paramters φ we expand V W Y ( G V V V... V V... γ I C... C γ I... G G ρ V 0 R(φ ρ V 0 R(φ... ρ V 0 R(φ ρ V 0 R(φ... γ G CG... C G γ G... Then the diagonal blocks are required to have the form V i i γ i G i ρ i V 0 i i R(φ i γ i ( βi α i βi A real solution requires that V i i > 0. We are free to choose the order of the conjugate columns of V to ensure that this is true. (Note that if we reverse the order of the columns V i,j V j,i, then the sign of the determinant of the X blocks is reversed. If we reverse the order of eigenvectors in V, then we also reverse the order of eigenvalues in D( θ or equivalently θ i π θ i. To find φ we proceed with our expansion of V 0 and R(φ i and write ρ i ( V0 V 0 V 0 V 0 ( ( cos φi sin φ i γ sin φ i cos φ i βi α i i βi We choose φ i so that G i 0, or tan φ i V 0 V 0 4

5 The ambiguity in φ i, (tan φ i tan(π φ i is resolved with the condition that G i V 0 cos φ i V 0 sin φ > 0. Summary. Find eigenvectors and eigenvalues. Transform eigenvectors to a real basis 3. Construct V. The columns of V are the eigenvectors. The eigenvectors appear as complex conjugate pairs since T is symplectic. 4. Choose the normalization for each pair of eigenvectors so that W will be symplectic. In particular if V c V, c V,... c V, c V,... where V i,j are X matrices, and c i ρe iφ i then choose ρ so that ρ ( V, + V, + V 3, Adjust the order of complex conjugate pairs so that V i,i > 0. That is, if V i,i < 0, than swap the order of the columns. 6. Choose the phases φ i so that has the form G i V i,i Rθ ( β 0 α β β 5