From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 2

Theorem 1.1 (Simultaneous diagonalization of two symmetric matrices).

n×n is a symmetric positive-definite matrix such
that the product AB − 1 exists, then there exists a non-singular matrix X such that both following
matrices are diagonal matrices, where I n is the n -dimensional unit matrix:

n×n is a symmetric matrix and B

If A ∈ R ∈ R

X T AX = diag( λ 1 ,...,λ n ) , X T BX = I n = diag(1 ,..., 1) .

(1.1)

End of Theorem.

According to our understanding, the theorem had been intuitively applied by C.F. Gauss when he
developed his theory of curvature of parameterized surfaces (two-dimensional Riemann manifold).
Here, the second fundamental form (Hesse matrix of second derivatives, symmetric matrix H)
had been analyzed with respect to the first fundamental form (a product of Jacobi matrices of
first derivatives, a symmetric and positive-definite matrix G). Equivalent to the simultaneous
diagonalization of a symmetric matrix H and a symmetric and positive-definite matrix G is the
general eigenvalue problem

| H − λ G | =0 , (1.2)

which corresponds to the special eigenvalue problem

| HG − 1 − λ I n | =0 ,

(1.3)

where HG − 1 defines the Gaussian curvature matrix

K=HG − 1 . (1.4)
In comparing two Riemann manifolds by a mapping from one (left) to the other (right), we here
only concentrate on the corresponding metric, the first fundamental forms of two parameterized
surfaces. A comparative analysis of the second and third fundamental forms of two parameterized
surfaces related by a mapping is given elsewhere. Uhlig ( 1979 ) published a historical survey of the
above theorem to which we refer. Generalizations to canonically factorize two symmetric matrices
A and B which are only definite (which are needed for mappings between pseudo-Riemann mani-
folds ) can be traced to Cardoso and Souloumiac ( 1996 ), ( Chu , 1991a , b ), Newcomb ( 1960 ), Rao and
Mitra ( 1971 ), Mitra and Rao ( 1968 ), Searle ( 1982 , pp. 312-316), Shougen and Shuqin ( 1991 ), and
Uhlig ( 1973 , 1976 , 1979 ). In mathematical cartography, the canonical formalism for the analysis of
deformations has been introduced by Tissot ( 1881 ). Note that there exists a beautiful variational
formulation of the simultaneous diagonalization of two symmetric matrices which motivates the
notation of eigenvalues as Lagrange multipliers λ and which is expressed by Corollary 1.2 .

−

Corollary 1.2 (Variational formulation, simultaneous diagonalization of two symmetric matrices).

n×n is a symmetric positive-definite matrix such
that the product AB − 1 exists, then there exist extremal (semi-)norm solutions of the Lagrange
function (tr[X T AX]) 1 / 2 =:

n×n is a symmetric matrix and B

If A ∈ R ∈ R

|| X || A , the A-weighted Frobenius norm of the non-singular matrix X

subject to the constraint

tr[X T BX − I n ]=0 , (1.5)

namely the constraint optimization

2
A

λ tr[X T BX

|| X || − − I n ]=extr X ,λ , (1.6)