Inverse transformation matrix entry bounds

[atrus_ovis]

I have sets of 2d vectors to be transformed by an augmented matrix A that performs an affine transform.
Matrix A can have values that differ at most |d| from the identity matrix, to limit the transformation, meaning that the min/max bounds for A are [itex] I_3 \pm dI_3[/itex]

The problem is that i'd lke to have bounds for the inverse as well, expressed as a function of d, so that if i know that the transformation matrix is bound by d, that the matrix of the inverse transformation is bound by f(d).
I thought the same bounds would apply, but they don't.
Is there a way to find them?

 

[atrus_ovis]

Nobody replies :(
Well i had the following idea:
make a matrix [ A , eye(3) , MinBound , MaxBound] , and reduce it to row echelon form.
That way the first 3x3 chunk will be eye(3) , the second the inverse of A , the third and fourth the respective min and max bounds for the inverse, hopefully.

 

[Stephen Tashi]

Matrix A can have values that differ at most |d| from the identity matrix, to limit the transformation, meaning that the min/max bounds for A are [itex] I_3 \pm dI_3[/itex]

Is d a scalar? Are you saying that A must be a diagonal matrix?

 

[chiro]

Hey atrus_ovis.

Try setting up the augmented system and find the inverse through row-reduction or by using co-factors (Cramers Rule) and you'll get an answer in terms of d.

 

[atrus_ovis]

Is d a scalar? Are you saying that A must be a diagonal matrix?

It is a scalar.A is not a diagonal,it's an identity matrix, where each value is shifted by at least/most +/- d.

Try setting up the augmented system and find the inverse through row-reduction or by using co-factors (Cramers Rule) and you'll get an answer in terms of d.

Yup,that's what i did.

 

[Stephen Tashi]

It is a scalar.A is not a diagonal,it's an identity matrix, where each value is shifted by at least/most +/- d.

The expression [itex] I_3 \pm d I_3 [/itex] only adds or subtracts d to the diagonal elements of the identity matrix [itex] I_3 [/itex].

 

[atrus_ovis]

You're right, i meant [itex] I_3 \pm \text{repmat(d,3,3)} [/itex] in matlabese.