On the Vandermonde Matrix Tom Copeland ([email protected]) May 2014 Part I The n-simplices and the Vandermonde matrix There are several denitions of the Vandermonde matrix Vn. Here 2 1 1 ··· 1 3 6 x1 x2 ··· xn 7 6 . 7 Vn = Vn(x1; x2; :::; xn) = 6 . .. 7 : 4 . 5 n−1 n−1 n−1 x1 x2 ··· xn with the determinant Y jVnj = jVn(x1; : : : ; xn)j = (xj − xi): 1≤i<j≤n The coordinates of the n vertices Cn−1(x1);Cn−1(x2); :::; Cn−1(xn) of the 2 n−1 (n − 1)-simplex along the unique moment curve Cn−1(x) = (x; x ; :::; x ) (ra- tional normal curve) that passes through the vertices comprise the columns of Vn (except for the rst row of ones). With one of the vertices located at the origin, say vertex m with xm = 0, the determinant gives the (possibly signed) hyper-volume of the hyper-parallelepiped framed by the position vectors from the origin to the other vertices, which bound the facet opposing the vertex at the origin (see the V 3 example below). Note that the xj are unit-less parameters, i.e., have no physical units or measures attached to them and in that sense are dimensionless. In other words, k should not be directly interpreted as a quantity with dimension a length, xj k an area, a volume, etc. It is a dimensionless parameter that must be interpreted in context. Ignoring this leads to confusion, such as believing that jV4j with six factors in its product formula must somehow have dimension six. The associated jVnj is also an eigenfunction of a generalized shift (dilation) operator with the face polynomial of the (n − 1)-simplex as the eigenvalue (cf. the MathOverow Q. A sum involving derivatives of Vandermonde?), i.e., 1 Pn k=1 exp[t· : xkDk :]jVnj = jVn((1 + t) · x1; x2; :::; xn)j + ::: + jVn(x1; x2; :::; (1 + t) · xn)j (1+t)n−1 , = [ t ] · jVnj where m m dm by denition for notational convenience and sug- : xkDk : = xk m dxk gestive parallelism with the form of the shift operator exp(tDk)f(xk) = f(t + xk). For example, look at the 3-simplex, the tetrahedron with face polynomial [(1 + t)4 − 1]=t = 4 + 6t + 4t2 + t3 (4 vertices/0-D faces,6 edges/1-D faces,4 triangles/2-D faces,1 tetrahedron/3-D face) and with the twisted cubic curve 2 3 as the parametrized moment curve with the th vertex at 2 3 (x; x ; x ) k (xk; xk; xk) along the curve (cf. MO-Q165283 The twisted kiss of the curvaceous cubic and the staid tetrahedron). The coordinates of n vertices of one (n − 1)-dimensional face, or facet, of the n-simplex along the moment curve comprise the moment matrix 2 3 x1 x2 ··· xn 2 2 2 6 x1 x2 ··· xn 7 6 . 7 Mn = Mn(x1; x2; :::; xn) = 6 . .. 7 4 . 5 n n n x1 x2 ··· xn The Vandermonde and moment matrices are related by Mn = Vn · diag(x1; x2; :::xn). Using this relation and the eigenvalue relation with t = −1, a recursion relation follows: n+1 jVn(x1; x2; :::; xn)j = jMn−1(x2; x3; :::; xn)|−|Mn−1(x1; x3; :::; xn)j+:::+(−1) jMn−1(x1; x2; :::; xn−1)j n+1 = (x2·x3 ··· xn)jV^n−1(x1)|−(x1·x3 ··· xn)jV^n−1(x2)j+:::+(−1) (x1·x2 ··· xn−1)jV^n−1(xn)j n+1 =e ^n−1(x1)jV^n−1(x1)j − e^n−1(x2)jV^n−1(x2)j + ::: + (−1) e^n−1(xn)jV^n−1(xn)j; with, e.g., jV^n−1(x2)j = jVn−1(x1; x3; :::; xn)j and e^n−1(x1) = en−1(x2; x3; :::; xn) = (x2 · x3 ··· xn), the highest order elementary symmetric polynomial for the (n − 1) indetermi- nates. The jMn−1j are the "hyper-volumes" of the hyper-parallelepipeds "framed" by the coordinate vectors of the (n − 1)-dimensional faces (facets) of the n - simplex. With one vertex placed at the origin (i.e., one xm = 0), only one jMn−1j is non-vanishing, giving the (possibly signed) hyper-volume of the hyper- parallelepiped framed by the position vectors to the vertices, which are those of the facet opposing the origin. 2 On the other hand, setting t = −1 in the eigenvalue equation for the deter- minants is equivalent to applying Cramers's rule to nd the rst column of the adjugate matrix of Vn, so dividing through by jVnj gives 1 n+1 1 = [^en−1(x1)jV^n−1(x1)|−e^n−1(x2)jV^n−1(x2)j+ :::+(−1) e^n−1(xn)jV^n−1(xn)j]; jVnj the inner product of the rst row of ones of Vn with the rst column of its inverse (see below for a 3-D example and the Appendix). Examples: - For V2(x1; x2), the associated 1-simplex is the line segment with vertices C1(x1) = x1 and C1(x2) = x2; therefore, jV2(x1; x2)j = jM1(x2)j − jM1(x1)j = x2 − x1; equaling a parametrized (unit-less) length, the distance between the 0- simplexes, the vertices C1(x1) and C1(x2). - For V3(x1; x2; x3), the associated 2-simplex is a triangle with vertices at 2 , 2 , and 2 ; therefore, C2(x1) = (x1; x1) C2(x2) = (x2; x2) C2(x3) = (x3; x3) jV3(x1; x2; x3)j = jM2(x2; x3)j − jM2(x1; x3)j + jM2(x1; x2)j; which equals the area of the parallelogram formed by the vectors to the endpoints of the 1-simplex, the side of the triangle from C2(x2) to C2(x3) minus that from C2(x1) to C2(x3) plus that from C2(x1) to C2(x2). 2 Consider x1 = −u, x2 = 0, and x3 = u > 0. Then jV3j = −|M2(x1; x3)j = 2u · u , the area ( u is a dimensionless parameter, i.e., it has no physical units) of the par- allelogram framed by the position vectors (−u; u2) and (u; u2) from the vertex 3 at the origin to the other two vertices, the endpoints of the opposing facet to the vertex at the origin. The product formula gives (u − 0)(u − (−u))(0 − (−u)) = 2u · u2, the base ( 2u) times the height ( u2) of the triangle. - For V4(x1; x2; x3; x4), the associated 3-simplex is a tetrahedron with ver- tices at 2 3 , 2 3 2 3 , and C3(x1) = (x1; x1; x1) C3(x2) = (x2; x2; x2);C3(x3) = (x3; x3; x3) 2 3 ; therefore, C3(x4) = (x4; x4; x4) jV4(x1; x2; x3; x4)j = jM3(x2; x3; x4)j − jM3(x1; x3; x4)j + jM3(x1; x2; x4)j − jM3(x1; x2; x3)j; which is the volume of the parallelepiped formed by vectors to the vertices of the 2-simplex, the triangle with vertices at C3(x2), C3(x3), and C4(x4) minus that from C3(x1), C3(x3), and C3(x4) plus that from C3(x1), C3(x2), and C3(x4) minus that from C3(x1), C3(x2), and C3(x3). Part II A recursive product formula from matrix inversion Looking at the inverse of V , i.e., nding the dual basis to V , leads to a re- cursive product formula. (The eigenvalue formula above actually encodes the column covectors of the inverse matrix. Compare the t = −1 example above with the inverse equation below and the Appendix. Also see MO-Q A sum involving derivatives of Vandermonde?.) The inversion process can be inter- preted geometrically as in the MO-Q Cavalieri's principle and the inversion of the Vandermonde matrix or in terms of orthonormality conditions of vectors and covectors. For example, for 3-D space spanned by the non-orthogonal column vectors of V3, I = N · S3 · V3 2 ^ 3 2 3 2 3 jV2(x1)j 0 0 e^2(x1) −e^1(x1) 1 1 1 1 1 ^ = 4 0 −|V2(x2)j 0 5 4 e^2(x2) −e^1(x2) 1 5 4 x1 x2 x3 5 jV3j 2 2 2 0 0 jV^2(x3)j e^2(x3) −e^1(x3) 1 x1 x2 x3 where I is the identity matrix, N is the rst normalizing diagonal matrix (incorporating the 1=jV3j), and S3 is the matrix of elementary symmetric poly- nomials, containing the non-normalized covectors to the column vectors of V3. A shorthand has been used as a mnemonic and for conciseness: jV^2(x1)j = det V2(x2; x3) = x3 −x2, jV^2(x2)j = det V2(x1; x3) = x3 −x1, and jV^2(x3)j = det V2(x1; x2) = x2−x1, similarly for the elementary symmetric poly- nomials, e.g., e^1(x2) = e1(x1; x3) = x1 + x3 and e^2(x2) = e2(x1; x3) = x1x3. 4 @(e1(x1;x2;x3);e2(:::);e3(:::)) S3 is essentially the transpose of the Jacobian (with @(x1;x2;x3) columns interchanged, mod signs) with jS3j = −|V3j, and since jIj = 1, it 2 3 follows that jNj = −1=jV3j , but also jNj = −|V^2(x1)j · jV^2(x2)j · jV^2(x3)j=jV3j ; therefore, jV^2(x1)j · jV^2(x2)j · jV^2(x3)j = jV3j: For higher dimensions, n−2 jV^n−1(x1)j · · · jV^n−1(xn)j = jVnj : You can easily convince yourself that the LHS of this formula contains (n=2) products of the factors of jVnj given that jVn−1j is given by the product formula. (Hint: use a table with individual dierences listed on the vertical and horizontal sides.) The relationships among the "coordinate systems" given by xi and ei, the characteristic polynomial, the Vandermonde matrix, and its inverse are clearly presented on pages 20 & 38 of "Hamiltonian 2-forms in Kahler geometry, I General Theory" by Apostolov, Calderbank, and Gauduchon.