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Results tagged with matrix-theory
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user 1898
Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to linear algebra). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur complements, structured and special matrices, matrix functions and equations.
2
votes
Accepted
Unique upper triangular basis matrix of sublattice $\Lambda \subseteq \mathbb{Z}^n$
Converted from a comment by another user:
I believe that this follows from the existence and uniqueness of Hermite normal form.
- 20.2k
0
votes
Recovering eigenvalues of a matrix from its $p$th compound matrix
Converted from a comment by another user:
Knowing the successive compound matrices of $A$ without knowing their
eigenvalues one can recover the elementary symmetric polynomials in
the eigenvalues of …
- 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
3
votes
Find a square, stochastic matrix of odd size, not a permutation matrix, with an eigenvalue o...
All examples in the answers given so far are block triangular with a permutation matrix as the diagonal block with the critical eigenvalue.
If you want something different, you can take
$$
M = \begin{ …
- 20.2k
4
votes
Solving linear matrix equation
This is just a system of 4 linear equations in the 4 unknown entries of $X$. Just write down those four equations and solve it. For generic $A,B,C$, it will be nonsingular, so there is going to be onl …
- 20.2k
1
vote
Accepted
Inequality for matrix with rows summing to 1
If I am not missing something, this seems a direct application of Titu's lemma
$$
\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
1
vote
Routh-Hurwitz criterion for matrices
The very boring answer, of course, is:
write down the characteristic polynomial $p(x) = \det(A-xI)$
write down the Routh-Hurwitz criterion for $p$, expanding everything in terms of the matrix coeffic …
- 20.2k
7
votes
Rank of $A\otimes B - B\otimes A$
A slight variant of Fedor's answer: using a QZ (generalized Schur) factorization $A=QT_A Z, B = Q T_BZ$, you can make an orthogonal change of basis such that $A$ and $B$ are both upper triangular. The …
- 20.2k
5
votes
General Sylvester's linear matrix equation
Can one expect a characterization similar to the Sylvester Theorem
As far as I know, no, apart from very special cases where the coefficients can be triangularized simultaneously. There is a big gap …
- 20.2k
3
votes
Accepted
For the purposes of solving linear equations, is there a fast decomposition that works for a...
From my comments: LDL variants that implement symmetric pivoting and avoid issues with zero diagonals have been invented in the 1970s: Bunch-Kaufman pivoting, Aasen's method for LTL factorization (the …
- 20.2k
5
votes
Accepted
Are there any results in generalizing matrix theory to multidimensional arrays?
Yes, many of them.
The keyword you should look for is tensors (note that it is used in a slightly different meaning in the physics literature, though).
I suggest to start from Kolda and Bader's 2009 S …
- 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
5
votes
Finding the nearest matrix with real eigenvalues
Vanni Noferini and I have published an arxiv e-print, Nearest $\Omega$-stable matrix via Riemannian optimization, in which we address also this problem (in Section 7.5).
Quick summary of our findings …
- 20.2k