# [SciPy-user] Inverse of complex matrix

Michael McNeil Forbes mforbes@physics.ubc...
Sat Nov 10 01:59:43 CST 2007

```Hi Roger,

Your matrix is extremely singular.  The singular values are:

>>> U,d,V = linalg.svd(R)
>>> d
array([1.41944451e+01,   1.23913727e+01,   9.46227279e+00,
2.81247753e-04,   1.11035378e-14,   1.02548448e-14,
8.22627497e-15,   6.74668324e-15,   5.51882714e-15,
4.78886237e-15,   2.82020737e-15,   7.14721524e-16])

Thus, the matrix is really il-conditioned (the ratio of the largest
to smallest singular value is of the order of the machine precision.
It is essentially singular, and only has 4 linearly independent
columns).   Thus, it is not surprising that the round-off errors are
significant.

You can't reasonably expect to invert a matrix when the ratio of the
singular values is of order the machine precision.

Michael.

On 9 Nov 2007, at 10:16 PM, Roger Herikstad wrote:

> Dear scipy users,
>  I am trying to invert a 12 by 12 complex matrix R using
> scipy.linalg.inv(), but the round-off errors seem a bit too big. If
> I do scipy.dot(Ri,R), the first row of the resulting matrix looks
> like this:
>
> array([ 0.953125  +0.j        ,  0.06542969+0.01171875j,
>         0.08154297+0.15234375j, -0.078125  -0.09375j   ,
>         0.19140625-0.0859375j ,  0.08203125+0.140625j  ,
>        -0.08398438+0.0078125j , -0.02539062-0.03125j   ,
>        -0.06738281+0.015625j  , -0.04736328+0.15820312j,
>        -0.09057617+0.02832031j,  0.109375  -0.0625j    ])
>
>
> Is this normal? Inverting a 2x2 matrix produces the expected
> result. I've attached a pickle of my matrix. Thanks!
>
> ~ Roger
> <matrix.pck>
>

```