An application of hyperintricate numbers –
the matrix symmetrizer classification
Jim Adams  2014
18th February 2014
Abstract. In 1910 Frobenius [Fr68] proved that any matrix is the product of two symmetric
ones. We seek to understand the classification of these symmetric matrices.
Note. When we refer to [Ad12], we will be concerned mainly with section 6 of that work, and
for [Ad13] mostly with section 11 of it.
Lemma. (a) [Ad12] If a matrix is symmetric the layers are composed each of basis element
of type i only, multiplied by a real factor, for an even number of layers, or they are of the
form a1 + c + d.
Proof. If an i layer was of the form a1 + bi + c + d, then for each J-abelian part in its
minimal summation, this could be expanded out to contain terms with an odd number of i’s,
which is antisymmetric. 
(b) If a matrix is antisymmetric, the layers are basis elements of type i multiplied by a real
factor for an odd number of layers, the remainder being of the form a1 + c + d.
Proof. Similar.
Lemma. Let W be a symmetric intricate number. Then W-1 is symmetric if it exists.
Proof. Let W = w1 + w + w, so
W-1 = (w1 – w – w)/(w2 – w2 – w2),
is symmetric. 
Lemma. Let W be a symmetric intricate and X and Y be arbitrary intricate. Then if XT is the
transpose of X, XTWX is symmetric.
Proof. Put X = x1 + xi + x + x. Let us compute
XTYX.
For the real part of Y, XTX is symmetric, since this is
(x2 + x2 + x2 + x2)1 + 2(xx + xx) + 2(xx – xx),
for the imaginary, i, part of Y, XTiX is
(x2 + x2 – x2 – x2)i,
for the actual, , part of Y XTX is
2(xx + xx)1 + (x2 – x2 + x2 – x2) + 2(xx – xx),
and for the phantom, , part of Y, XTX is
2(xx + xx)1 + 2(xx – xx) + (x2 – x2 – x2 + x2). 
Corollary. Let A and D be symmetric intricate numbers and C = BT. Then what is known
from [Ad13] as the inverse of the Schur complement
Sc = (D – CA-1B)-1
is symmetric. 
Lemma. Let M be a general symmetric matrix and X be an arbitrary matrix. Then if XT is
the transpose of X, XTMX is symmetric.
Proof. Any m  m matrix M may be extended by the introduction of unit diagonal elements,
the remainder of the elements in the extension part being zero, to a 2n  2n n-hyperintricate
matrix. Consider a symmetric matrix M and extend it to a matrix L in the aforementioned
manner.
This lemma follows as a generalisation, under a hyperintricate representation, of the previous
lemma, in which Y may contain an even number of purely i layers to make it symmetric. 
Theorem. (Frobenius, 1910). Let M, M be m  m symmetric matrices to be determined.
Then any m  m matrix P may be represented by MM.
We will first establish this theorem in the case m = 2.
Proof. To start off the induction in effect for the general case, let n = 1,
P = q1 + ri + t + u,
and
L = a1 + c + d.
Then
L-1 = (a1 – c – d)/(a2 – c2 – d2),
and
P = (PL-1)L,
where PL-1 is symmetric, that is
PL-1 = [(qa – tc – ud)1 – (td – uc)i – rd + rc]/(a2 – c2 – d2)
is symmetric, so td – uc = 0 and (a2 – c2 – d2)  0. 
We now consider the case n = 2.
A B
Theorem. Say L = [
] is symmetric, where A, B, C and D are (n – 1)-hyperintricate
C D
matrices, then A and D are symmetric, and B is the transpose of C:
B = CT,
with L-1 represented in the case n = 2 as follows.
E F
From [Ad13], L-1 = [
], where
G H
E = A-1 + A-1BScBTA-1,
H = S c,
and since A-1 and Sc are symmetric
A-1 = A-1T,
S c = S cT
so that by the lemma E and H are symmetric, whereas
F = -A-1BSc,
G = -ScBTA-1. 
We will represent a general 2n  2n hyperintricate matrix by
Q R
V=[
],
T U
so that we wish to prove that there exists a
V = (VL-1)L,
QE + RG QF + RH
VL-1 = [
],
TE + UG TF + UH
where VL-1 is symmetric under a suitable configuration of L. What are the constraints on E,
F, H and G, so this holds for arbitrary Q, R, T and U?
Then in the case n = 2, A and D, being diagonal, have no i layer, whereas if B has no i layer
then nor has C, and if B has an i layer component, then for C the layer component is –i (the
transpose iT of i is –i).
References
Ad12 J.H. Adams, Introduction to intricate and hyperintricate numbers I, 2012, to be found
in Innovation in mathematics, online at www.jimhadams.com, 2014.
Ad13 J.H. Adams, Introduction to intricate and hyperintricate numbers II, 2012, to be
found in Innovation in mathematics, online at www.jimhadams.com, 2014.
Fr68 G.F. Frobenius, 81. Über die mit einer Matrix vertaushbaren Matrizen, Gesammelte
Abhadlungen, vol III, Springer, 1968.