Inverse Matrix

[rictheric222]

I have an assignment in c programming where I must calculate the inverse of a given matrix.
The formula that was given to me is A^-1 = I + B + B^2 + B^3 + ... (directed to stop at B^20) where B = I - A
Does this make sense?
I'm having a hard time understanding this for some reason... I feel like the result would end in inf's and -inf's...
If someone could explain this concept to me, that would be awesome.

Thanks in advance.

 

[Romsek]

I have an assignment in c programming where I must calculate the inverse of a given matrix.
The formula that was given to me is A^-1 = I + B + B^2 + B^3 + ... (directed to stop at B^20) where B = I - A
Does this make sense?
I'm having a hard time understanding this for some reason... I feel like the result would end in inf's and -inf's...
If someone could explain this concept to me, that would be awesome.

Thanks in advance.

this comes from the properties of the geometric series of a matrix.

If A is a an invertible matrix with all eigenvalues |lambda_i| < 1 then the following holds true

SUM[0,Infinity,Aⁿ] = I/(I-A) = (I-A)^-1

consider A^-1 = (I - (I-A))^-1

here we have the (I-A) playing the role of the original A and so

SUM[0,Infinity,(I-A)ⁿ] = (I-(I-A))^-1 = A^-1

and this converges if (I-A) has all eigenvalues |lambda_i| < 1
The fact that the eigenvalue magnitudes are all under 1 keeps the series from blowing up as you suspected. If this isn't true the series will indeed tend to infinity.