Help with Gram-Schmidt Orthogonalization I am reading this paper. It is about inequality constraint in Kalman filter.
http://repository.cmu.edu/cgi/viewcontent.cgi?article=1748&context=robotics
and come across this matrix equation.
$$VW^{\frac{1}{2}}T^T\phi_i = [(\phi_i^TC_{i-1}\phi_i)^{\frac{1}{2}}\quad 0\ \cdots\quad 0]$$
where $T$ and $W$ can be calculated through Jordan canonical decomposition ($C = TWT^T$ and $C\in\mathbb{R}^{n\times n}$ is known) and $V$ is orthogonal matrix which can be obtained by using Gram-Schmidt orthogonalization. The vector $\phi_i\in\mathbb{R}^n$ is also known. I have two questions.


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*Can I use eigenvalue decomposition instead of Jordan canonical decomposition?  I tried both of them in MATLAB, the results are quite similar except that all of eigenvectors are divided by last element in Jordan decomposition. And Jordan decomposition is very expensive.

*How to find matrix $V$? Paper and all of references tell me to find matrix $V$ by using Gram-Schmidt orthogonalization, but what would be its argument?

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*My guess is that the reason Jordan form is used it because the eigenvalue decomposition doesn't always work if applied naively. For example, for the case where the geometric multiplicity of a repeated eigenvalue does not match it's algebraic multiplicity. How to find the multiplicity of eigenvalues?

*I think the argument of the orthogonalization are the vectors $\phi_{i}$. You can use the QR decomposition for this. In other words, do the QR decomposition on $VW^{1/2}T^{\rm T}\phi_{i}$. Then the matrix $Q^{\rm T}$ is $V$. Note: if $\phi_{i}$ is a column vector, then T is just 1.
Hope this helps.
