Inverse of symmetric 3x3 matrix
Author
Given a symmetric 3x3 matrix. Are there any properties of a symmetric 3x3 matrix that I can use in order to accelerate the inversion? Is there a faster way for inverting a symmetric 3x3 matrix opposed to the normal inversion methods (e.g. cofactor expansion, Cramer, ...) or can I use properties of the symmetric matrix in these methods in order to improve the inversion?
-Dirk
Well, if you use Cramer's rule for calculating the inverse, you'll find a lot of common subexpressions that can be factored out given a symmetric matrix.
Let this be the matrix:
Its determinant is: D = -m11*m12^2+ m11^2*m22 - m13^2*m22 + 2*m12*m13*m23 - m11* m23^2
If it is non zero, proceed.
The members of the inverse:
a11 = m11*m22 - m23^2
a12 = m13*m23 - m11*m12
a13 = m12*m23 - m13*m22
a22 = m11^2 - m13^2
a23 = m12*m13 - m11*m23
a33 = m11*m22 - m12^2
Divide them all by the determinant.
The rest three are equal to their symmetric ones, since the inverse will be symmetric too.
I hope I copied them correctly, let me know if something is wrong.
[Edited by - someusername on March 9, 2006 11:00:09 AM]
[m11 m12 m13][m12 m22 m23][m13 m23 m11]Its determinant is: D = -m11*m12^2+ m11^2*m22 - m13^2*m22 + 2*m12*m13*m23 - m11* m23^2
If it is non zero, proceed.
The members of the inverse:
a11 = m11*m22 - m23^2
a12 = m13*m23 - m11*m12
a13 = m12*m23 - m13*m22
a22 = m11^2 - m13^2
a23 = m12*m13 - m11*m23
a33 = m11*m22 - m12^2
Divide them all by the determinant.
The rest three are equal to their symmetric ones, since the inverse will be symmetric too.
I hope I copied them correctly, let me know if something is wrong.
[Edited by - someusername on March 9, 2006 11:00:09 AM]
Author
Silly me. I derived this as well today but totally missed that the inverse is symmetric as well. Thanks for this.
-Dirk
-Dirk
Well, (M-1)T = (MT)-1 for all square matrices M.
If one happens to be equal to its transpose (symmetric), then
M = MT <=>
M-1 = (MT)-1 = (M-1)T
thus M-1 is symmetric too.
I don't see another property that can be used to speed up things more...
If one happens to be equal to its transpose (symmetric), then
M = MT <=>
M-1 = (MT)-1 = (M-1)T
thus M-1 is symmetric too.
I don't see another property that can be used to speed up things more...
I can't recall any special methods for inverting a symmetric matrix.
If one existed I'd venture to guess that since all symmetric matrices are hamiltonian matrices, you could use a Jacobi transformation to diagonalize it then inverting the diagonalized matrix is simple. Dunno if that is any better than the regular inversion methods, though.
If one existed I'd venture to guess that since all symmetric matrices are hamiltonian matrices, you could use a Jacobi transformation to diagonalize it then inverting the diagonalized matrix is simple. Dunno if that is any better than the regular inversion methods, though.
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