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.
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?
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.