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1 vote

Questions on symmetric Hadamard matrices

Currently, this is an open problem. So I only got a partial answer. We just need to consider the following two different situations: Case 1: $n$ is not a perfect square. There is no solution if $\...
user369335's user avatar
5 votes
Accepted

Forming real positive semidefinite matrices from complex matrices

We say that a matrix $O$ is complex orthogonal if $O^TO=I$ where $I$ denotes the identity function. If $n>1$, then there are many complex orthogonal matrices that are not real orthogonal matrices. ...
Joseph Van Name's user avatar
4 votes

Unitary transformations of Vandermonde matrices forms a smooth manifold?

The answer is 'not always', in particular, not when $(n,r)=(2,3)$. The image is obviously is a smooth manifold when $n=0$, for then the image in $\mathbb{R}^{(n+1)r}=\mathbb{R}^r$ is the sphere $\...
Robert Bryant's user avatar
0 votes

Generate a low-rank sparse covariance matrix

Matlab's sprandsym generates a random sparse positive-definite matrix by starting from a diagonal matrix and applying to it Jacobi rotations, i.e., rotation ...
Federico Poloni's user avatar
2 votes
Accepted

Where does $V$ from the spectral decomposition $A = VDV^*$ lie, if $A$ has only imaginary entries?

The $n\times n$ imaginary matrix $A$ satisfies $A^\top=-A$, so it is skew-symmetric. The Youla decomposition is $$A=iO\Sigma O^\top,$$ where $O$ is a real orthogonal matrix and $\Sigma$ is a real ...
Carlo Beenakker's user avatar

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