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Results tagged with matrix-analysis
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user 1898
The study of the properties of real and complex matrices that are more close to analysis and operator theory. For instance: the properties of positive definite matrices, matrix inequalities, perturbation analysis, matrix functions, inequalities between eigenvectors and singular values, majorization.
3
votes
Under what conditions does $x^TA^{-1}y> 0$ hold? $A$ is a symmetric positive definite matrix...
Just a general comment: you might be interested in checking out the theory of M-matrices. An M-matrix is a matrix such that $M_{ij} \leq 0$ for $i\neq j$, with the additional property that all its eig …
- 20.2k
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 matrices that act only on two components. I …
- 20.2k
3
votes
How expressive is $e^A$ in the sense of universal approximation?
[EDIT: added and then removed a stronger argument that did not work.]
A partial answer providing a starting point and expanding on the comment:
if $B$ has no real negative eigenvalues, the answer is …
- 20.2k
2
votes
Singularity of matrix pencil-like expression
The statement is false.
Take $3\times 3$ matrices such that $A_{11}=B_{22}=1$ and all other entries are zero. Then $EA-hB$ has the third column equal to $0$, but the row spaces of $A$ and $B$ are disj …
- 20.2k
1
vote
Accepted
Inequality for matrix with rows summing to 1
If I am not missing something, this seems a direct application of Titu's lemma
$$
\sum_{k=1}^K \frac{x_k^2}{y_k} \geq \frac{\left(\sum_{k=1}^K x_k \right)^2}{\sum_{k=1}^K y_k}, \quad x_k \geq 0, y_k > …
- 20.2k
2
votes
Accepted
Limitation through the singular values
Yes, one can prove that
$$
|\sigma_i(A+E) - \sigma_i(A)| \leq \|E\| \quad \quad \forall i
$$
which implies that the singular values are continuous. This follows, for instance, from Weyl's inequalities …
- 20.2k
3
votes
Accepted
Is there a specific name for this optimization problem?
It's the symmetric version of the low-rank approximation problem.
- 20.2k
3
votes
How do you solve this quadratic matrix equation?
This question is not an exact duplicate, but my answer gives you a pointer that applies also to your case.
- 20.2k
2
votes
How could I extend this result to a case where the matrices were not of full rank?
The natural way to relax the condition $A \geq XB^{-1}X^*$ to the case when $B$ is not full-rank is transforming it to
$$
\begin{bmatrix}
A & X\\
X^* & B
\end{bmatrix} \geq 0,
$$
which is equivalent b …
- 20.2k
3
votes
Properties of matrix $X=\left[\frac{1}{1-\bar\alpha_i \alpha_j}\right]_{ij}$
Assuming all the $\alpha_j$ are nonzero, the matrices $X$ are Cauchy-like matrices, since you can rewrite them as
$$
X_{ij} = \frac{\alpha_j^{-1}}{\alpha_j^{-1}-\bar{\alpha}_i}
$$
so there are analogo …
- 20.2k
17
votes
Accepted
Closed form solution for $XAX^{T}=B$
$B^{-1/2}XAX^TB^{-1/2}=I$, so $B^{-1/2}XA^{1/2}=Q$ must be orthogonal. On the other hand, for any orthogonal $Q$, it is simple to verify that $X = B^{1/2}QA^{-1/2}$ solves the equation, so this is a c …
- 20.2k
1
vote
Accepted
A question of invertibility of matrices
There is a Jordan-like canonical form for symmetric matrix pairs $(A,B) = (A^*,B^*) \in \mathbb{C}^{n\times n} \times \mathbb{C}^{n\times n}$ under the transformation $(A,B) \to (M^*AM,M^*BM)$, with $ …
- 20.2k
2
votes
Accepted
Reference request: continuity of Cholesky factor
A subtle issue is that $\Pi$ is not unique here. For instance, if
$$
A = \begin{bmatrix}
1 & 0 & 0\\\\
0 & 0 & 0\\\\
0 & 0 & 0
\end{bmatrix}
$$
then you can take both the identity and $(23)$ as the …
- 20.2k
2
votes
Comparison of methods to define a matrix function (Jordan canonical form, Hermite interpolat...
None: they are all equivalent and all three return the same result for any function that's defined on the spectrum of $A$. (Theorem 1.12 in Higham's book Matrix Functions.) The only minor drawback of …
- 20.2k
1
vote
Accepted
A "positive diagonal plus skew-symmetric" matrix decomposition
Choose any positive definite matrix $Q$. Since $A$ has eigenvalues with positive real part, the Lyapunov equation $$AP + PA^\top = Q$$
is solvable, and its solution $P$ is symmetric and positive defin …
- 20.2k