Review of A Course in the Geometry of n Dimensions
By Steve Hellinger
20 May 2008.
I have recently worked my way through the entire monograph and I found a large number of issues that potential readers should be aware of.
Many mistakes in the statement of equations and relations
In Kendall's Example 4 on page 10 he seeks to derive the number of r-boundaries of an n-dimensional parallelotope. In equation (4) he defines a generating function with one associated condition N0 = 1. This condition is incorrect, we actually should have
Nn = 1[ In fact N0 is 2n ]. For the recursion relation that immediately follows (4) we will need the unstated conditions N'-1 = 0and N'n-1 = 1.Finally in the relation of sums that immediately precedes equation (5) the upper limit of the sum on the right-hand side of the equal sign should be n-1, not n.
Inconsistent and confusing use of notation from one section to the next, and even from one paragraph to the next
In sections 28 and 29 (pp. 22-27) Kendall considers the angles between flats other than S1 and Sn-1 in Sn. On page 23 (in section 28) he defines a Lagrange multiplier lambda1 as a (1×p) vector (i.e., a row vector). But later in the same derivation, in equation (73) and the argument immediately following it, the same lambda1 has become a (p×1) vector (i.e., a column vector). There was no indication to the reader that the change was to be made.
A serious flaw in a major proof
Again in sections 28 and 29 (pp. 22-27), where Kendall considers the angles between flats other than S1 and Sn-1 in Sn, he seeks to prove that the lines corresponding to the minimal angles between the flats are orthogonal in their respective spaces. He defines a matrix K that is the product of two symmetric matrices and then states that K is symmetric. However K is not symmetric under the general conditions assumed by Kendall for its component matrices, and those component matrices are derived from an arbitrary (p×n) matrix A and an arbitrary (q×n) matrix B. Further, this lack of symmetry can be confirmed in Kendall's Example 8 on page 26. The elements of the 2×2 matrix K in Example 8 can be computed to be:
k1,1 = (32 + 1/2)/51, k1,2 = 1 / (2 × 51), k2,1 = 1/2, k2,2 = 1/2
and thus K in Example 8 is clearly not symmetric.
Kendall's proof of the orthogonality condition described above requires that K be symmetric (so that eigenvectorsassociated with distinct eigenvalues of K will be orthogonal), hence his proof is invalid.
We can rescue Kendall's proof by returning to his use of the arbitrary (p×n) matrix A in equation (47):
AL = 0
where L is an (n×1) column vector that represents a line in one of the flats. We recognize that (47) states that the vector L must be orthogonal to the row space of the matrix A. So we may apply Gram-Schmidt orthogonalization to the row vectors of A and replace them with a set of mutually orthogonal row vectors, and make each of them the same length (the common length doesn't have to be unity). The vectors L that solve (47) will be the same as those that solve our modified matrix equation. Then we do the same procedure for the row vectors of the arbitrary (q×n) matrix B that appears in equation (49):
BM = 0
where M is an (n×1) column vector that represents a line in the other flat.
In these modified circumstances the derived matrix K is symmetric and Kendall's proof is valid.
Inadequate definitions of terms in quite a few places
In equation (85) on page 29 Kendall presents a formula for the quadric Vn-1. The listed bi,j should in fact be bi. Moreover Kendall's definition that follows
bi, n+1 = ai, n+1
is inadequate, we must have
bi = an+1, i + ai, n+1
Very numerous typographical errors
I have found typos on 21 pages (of 63 total) and some pages have several typos. Some examples follow:
Equation (33) on page 19: In the last row of the determinant xn,1 and xn,2 should be x1,n and x2,n respectively.
Equation (157) on page 53: The second "S" summation symbol on the right-hand side of the equal sign should not be there.
Equations (158) and (160) on page 53: The factor np/2 in the denominator should not be there. Its erroneous presence can be traced to an improper accounting for the Helmert transformation with respect to the volume element.
Equation (162) on page 54: The subscript on the left hand side of the equation should be mi, not mj.
A lack of references to the works from which much of the statistical section is drawn
Section 52, "Student's" t, is based upon William S. Gosset's original 1908 paper (published under the pseudonym "Student") and later mathematical confirmation using geometric arguments by R.A. Fisher.
Section 53, "The mean in rectangular variation", is based upon Hall (1927).
Section 54, "The correlation coefficient in bivariate normal variation", is based upon Fisher (1915).
Section 55, "Wishart's distribution", is based upon Wishart (1928).
Fisher, Ronald A., 1915, Frequency Distribution of the Values of the Correlation Coefficient in Samples from an Indefinitely Large Population, Biometrika, Vol. 10, No. 4 (May 1915), pp. 507-521.
Hall, Philip, 1927, The Distribution of Means for Samples of Size N Drawn from a Population in which the Variate Takes Values Between 0 and 1, Biometrika, Vol. 19, No. 3/4 (Dec 1927), pp. 240-245.
Student (William S. Gosset), 1908, The Probable Error of a Mean, Biometrika, Vol. 6, No. 1 (March 1908), pp. 1-25.
Wishart, John, 1928, The Generalised Product Moment Distribution in Samples from a Normal Multivariate Population, Biometrika, Vol. 20A, No. 1/2 (July 1928), pp. 32-52.
A Course in the Geometry of n Dimensions, by M. G. Kendall. Dover Publications, 2004. Softcover, 63 pp., $7.95. ISBN 0-486-43927-5.