# Tagged Questions

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### How do you generate orthogonal Rotations in d-dimensions? [on hold]

I know that matrices that satisfy $\mathbf{x^TMx} > 0$ for all $\mathbf{x} \in \mathbb{R}^d$ are called positive definite matrices. It seems to me that a matrix that represents an $90^{\circ}$ ...
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### Proving that the eigenvalues of a certain matrix product are positive

Let $A$ be an $m \times n$ matrix, and define: \begin{align*} U &= {\rm diag} \{ \frac{1}{\beta_j} \}, \beta_j = \sum_{k=1}^m |a_{kj}|, j = 1 \dots n \\ V &= {\rm diag} \{ \frac{1}{\alpha_i} ...
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### Matrix, singular values, Moore-Penrose-pseudoinverse

If A is any real mxn-matrix consider the block matrix $\begin{pmatrix} E&A^T \\ A&0\end{pmatrix}$. This matrix seems to have close connections with pseudo inverse, svd etc. which are probably ...
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### Sensitivity of the range of a matrix

The distance between two subspaces $\mathcal{U}$ and $\widetilde{\mathcal{U}}$ is classically defined as $d(\mathcal{U},\tilde{\mathcal{U}}):=\|P-\tilde{P}\|$, where $P$ and $\tilde{P}$ are orthogonal ...
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### Inverses of the sums of all possible subsets of a set of symmetric and positive definite matrices

I have a set of $c$ matrices $A_1 ... A_c$ which are all symmetric and positive definite. I would like to calculate the inverses of all the possible sums, i.e. ...
Let $\Sigma$ be a correlation matrix, ie. symmetric. The Choleski decompositon gives upper triangular $A$ such that $A^TA = \Sigma$. Instead of upper triangularity, we are looking for $A$ that is not ...