MATLAB: Multiplying by inverse of a matrix

inverse multiplication

Hello,
I want to calculate where α is a scalar (it's for calculating the estimated variance of a parameter). Using
alpha*inv(X'X)
gives the correct results but (a) Matlab suggest not doing so (although the backward slash gives the wrong results) and (b) I've always avoided multiplying by the inverse of a matrix due to potential inaccuracy.
Is there a better way?
Thank you

Best Answer

You are essentially "dividing" by the X'*X quantity, so that is what needs to appear on the "bottom" of the slash. E.g.,
>> alpha = 0.5;
>> X = [0.6 0.9; 0.9 0.5];
>> alpha*inv(X'*X)
ans =
    2.0377   -1.9031
   -1.9031    2.2491
>> (X'*X)\eye(2)*alpha
ans =
    2.0377   -1.9031
   -1.9031    2.2491
>> alpha*eye(2)/(X'*X)
ans =
    2.0377   -1.9031
   -1.9031    2.2491
Or you can think of it this way. Start with this definition:
inv(X'*X) * (X'*X) = eye(2)
and solve for the inverse:
inv(X'*X) = eye(2)/(X'*X)
Similarly, starting with this definition:
(X'*X) * inv(X'*X) = eye(2)
yields
inv(X'*X) = (X'*X)\eye(2)
Then just multiply by alpha.
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