# Do you know this matrix?

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Hi All, Here's a matrix with elements A-I:
M =[
A B C
D E F
G H I
]
So, a bit of a long shot here, but does anyone then recognise the following symmetrical 4x4 matrix:
N = [
A+E+I F-H   G-C    B-D
F-H   A-E-I B+D    G+C
G-C   B+D   -A+E-I F+H
B-D   G+C   F+H    -A-E+I
]
( BTW the sum of its diagonal elements is zero. ) Any hints at all would be gratefully received. Thanks in advance, Graham [Edited by - Fruny on August 30, 2005 10:08:29 AM]

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Where did you get that matrix from? What do you mean by "recognize"?
A bit of context would help.

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A context would help a great deal. It looks familar, but without a context I can't even guess where I saw it at. One observation is that zero maps to zero, but nothing maps to identity.

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My apologies, I should have given more context. It's from Berthold Horn's paper on absolute orientation (See Page 7 of http://people.csail.mit.edu/bkph/papers/Absolute_Orientation.pdf) Although I have made full practical use of this paper, I'd like to, but can't follow its derivation of matrix N.

I'm hoping someone may recognise the pattern of elements in N, perhaps from another area of mathematics where I may get a second chance to understand it.

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I'm afraid it was just a vague similarity to other things rather than a vague memory of this particular matrix. The basic layout of the matrix is similar to a matrix representation of a quaternion. The main diagonal is similar to the components of the product of two quaternions. My knowledge of quaternions is extremely limited. My only experience with them is as a type of number in abstract algebra. We did a section on relating quaternions represented as 4x4 matrices to 3x3 orthonormal matrices, but I can't find the notes.

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Look for hermitian matrix and pauli spins it might give a clue where it is derived from.

[Edited by - Name_Unknown on September 1, 2005 12:19:04 AM]

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I briefly looked over the paper you linked. It appears that N is derived in the section immediately preceding the one where M and N are written out. N is first derived as a summation of matrix products.

The section describing M and N apparently constructs M as another way to explain the matrix N and how it works, but it doesn't seem that N is directly derived from M, rather it is a different way of viewing the matrix.

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