Multiplying by inverse of a matrix

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

[J. Alex Lee]

can you give example values of alpha and X?

[JPF]

For example,

alpha = 0.5;
X = [0.6 0.9; 0.9 0.5];

For this I obtain

alpha\(X'*X) = [2.34 2; 2 2.2]
alpha*inv(X'*X) = [2.04 -1.9; -1.9 2.2]

Answer 1

[James Tursa]

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.

[JPF]

That's great, thank you. I was thrown off by Matlab's recommendation that I just used the backward slash.