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What do you mean by "positive linear combination"? Try G=I. If you select any column of G, you'll find that its projection onto the space spanned by the other columns of G is the 0 vector.

I'm not aware of anything about $B^{+}\_{s}b_{c}$ that would be helpful here. It's not a column of the identity, because $B^{+}B$ isn't generally $I$. You can easily confirm this by looking at a ran …

This certainly won't work when $A=0$ and $b\neq 0$. In that case, $Ax=b$ has no solution and $A^{T}y>0$ has no solution. Another counterexample to consider is $A=[1\;\; -1; -1\;\; 1]$ and $b=[2; 1 …

In practice using algorithms in EISPACK or LAPACK on floating point single or double precision symmetric matrices, computing the EVD takes $O(n^3)$ time. The constant hidden within the big $O$ is con …

In this case, it's easy enough to compute the probability exactly in terms of the incomplete beta function. Let $v$ be the fixed unit vector, and $r$ be the random unit vector, uniformly distributed …

No, that's not quite the generalization that you'll get when you extend the Schur complement theorem for positive definite matrices to negative definite matrices. Try using the fact that a matrix $X$ …

In practice, on typical desktop computers and server class machines using the x86-64 architecture, matrix-vector multiplication is limited more by memory bandwidth than floating point operations. Thi …

You haven't said so, but I'm assuming that $\psi$ and $\phi$ are vectors. These could more generally be functions in some function space, and you would typically discretize those functions to work wi …

There's a very large literature on updating solutions to least squares problems as new data are added. The naive formula $\hat{\beta}=(X^{T}X)^{-1}X^{T}y$ can be problematic in practice because of nu …

If $Ax=b$ has a unique solution $x^{*}$, then $x_{m}=x^{*}$. If $Ax=b$ has infinitely many solutions, then $x_{m}$ will be one of these solutions. In particular, it will be the solution with the s …

Algorithmically, you can solve the linear programming problem: $\max \sum_{i=1}^{n} x_{i} $ $Ax=0$ $x \geq 0$ If the maximum is strictly greater than zero, then there's a nonzero solution that sa …

You haven't said whether you're interested in theoretical analysis or practical computation. At the practical level, you also haven't specified whether you can live with an approximate floating poin …

I agree that this is a better question for scicomp.stackexchange.com. Maybe. It depends on the sparsity structure of your particular matrix and the actual numerical values of the nonzero elements …

I'm assuming that by $\| A^{1/2}b-z \|$, you're referring to the 2-norm and that by $A^{1/2}$, you're referring to the unique symmetric matrix square root of $A$. If you can precompute $A^{1/2}z$, …

"Covariance Matrix" doesn't make a lot of sense in this situation, since there's no multivariate normal distribution around the fitted parameters. You could if you wanted look at a non-ellipsoidal co …