Constructions of harmonic maps between Hadamard manifolds

2.1. EIGENMAPS 45
Following a method due to M. Parker [30], who applied it to the case m = n, we now
define an mn × mn-matrix G(F )by
⎡
⎤
I n A 12 ··· A 1m
A
G(F ):=
21 I n ··· A 2m
,
⎢ . . . ⎥
⎣
⎦
A m1 A m2 ··· I n
where I n denotes the n × n identity matrix and A ij are n × n-matrix whose entries are
(A ij ) kl = 〈a ik ,a jl 〉, 1 ≤ k, l ≤ n.
By virtue of (2.1.1), each A ij is a skew-symmetric matrix and A ji = −A ij . Note that the
determinant of G(F ) coincides with Gram’s determinant with respect to the system of vectors
{a ij }. Hence rank G(F )=r.
We consider only the case of m =2, and prove the following existence result of orthogonal
multiplications.
Proposition 2.1.2. There exists a full orthogonal multiplication F : R 2 × R n → R r if and
only if r is even, where n ≤ r ≤ 2n.
Proof. We first prove that rank G(F )(= r) must be even whenever a full orthogonal
multiplication exists. Recall that the characteristic polynomial of G(F )is
⎡
⎤
det(G(F ) − µI 2n ) = det ⎣ (1 − µ)I n −A
⎦ ,
A (1 − µ)I n
where A = A 21 . Since
⎡
det ⎣ A
B
−B
A
⎤
⎦ = ∣ ∣ det[A +
√
−1B]
∣ ∣
2
,
A and B being real matrices and |·|denoting the absolute value, we have
det(G(F ) − µI 2n )= ∣ ∣det[(1 − µ)I n + √ −1A] ∣ ∣ 2 .