Why is the inverse of a symmetric matrix not symmetric?!

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Ted
Ted 2013-5-8
Hi all, As far as I know, the inverse of symmetric matrix is always symmetric. However, I have a symmetric covariance matrix, call it C, and when I invert it (below), the solution, invC, is not symmetric!
>> invC = inv(C); % (inefficient I know, but it should still work...)
>> isequal(invC,invC')
ans = 0
Has anyone had this issue? Can this be due to rounding errors? My matrix is 1810x1810 with many entries like 0.0055, etc.
Thanks in advance!

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采纳的回答

Roger Stafford
Roger Stafford 2013-5-8
Yes, it's roundoff error. Instead of 'isequal' which demands exact equality, try displaying the difference invC-invC' to see if the differences fall within the range of what you would regard as reasonable round off errors. With a matrix which is close to being singular these can be surprisingly large sometimes.
  2 个评论
Ted
Ted 2013-5-10
Thanks for the tip.
The errors were small. I've now forced my "inverse" to be symmetric:
>> invCtrans = invC';
>> invC = triu(invC) + tril(invCtrans) - diag(diag(invC);
What I'm really trying to do now is use the chol() command for cholesky factorization. I know I need a symmetric positive definite matrix (spd), and I've checked by using eigs(invC,10,0), which tells me the 10 smallest eigenvalues are all positive. I'm still getting the following though:
>> U = chol(invC)
error using chol: matrix must be positive definite
... Any thoughts? Doesn't my eigenvalues test show that invC is spd?
Ted
Ted 2013-5-10
Never mind. I was getting the 10 eigenvalues with smallest magnitude, rather than the "most-negative."

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更多回答（1 个）

Youssef Khmou
Youssef Khmou 2013-5-8
编辑：Youssef Khmou 2013-5-8
hi,
Try to use a tolerance criterion :
C=symdec(100,100);
C=C/max(C(:))
I1=inv(C);
I2=inv(C');
norm(I1-I2) % its not zeros but saturated to zero (1e-n , n>20 )

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