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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.
8
votes
Cholesky decomposition of a positive semi-definite
You can either:
use a LDL^T decomposition (see e.g. here)
deflate the kernel yourself before: that is, compute a basis $Q_2$ for the kernel, complete it to a square orthonormal matrix $Q=[Q_1 \, Q …
- 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
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 …
11
votes
Accepted
When the sum of positive definite matrices converges, does the sum of the norm of the associ...
You can bound $\|A_k\| \leq C(n)\max_{i,j} |(A_k)_{ij}|$ for some function of the dimension only $C(n)$, because all norms are equivalent in finite dimension. If I am not mistaken $C(n)=\sqrt{n}$, but …
- 20.2k
1
vote
Accepted
zero patterns of M-matrices and their invereses
I haven't encountered this exact result, but a closely related one is a complete characterization of the possible zero patterns for inverse M-matrices in https://doi.org/10.1016/j.laa.2009.03.022.
It …
- 20.2k
1
vote
A conjugation matrix $X\in \mathbb{C}^{ n\times p}$ where $p< n$
Let $A=R^*R, B=S^*S$ be Cholesky factorizations. Then for any orthogonal $Q\in\mathbb{C}^{n\times n}$ the matrix $X=R^{-1}Q\begin{bmatrix}S\\0\end{bmatrix}$ works, as can be verified directly.
- 20.2k
1
vote
If $S$ is a nonsingular symmetric matrix over a number field and $D_k$ is its principal mino...
That seems to be a mistake. My guess is that the $\neq 0$ should have been $>0$ instead; then the result should hold.
(I'm posting this as an answer to prevent this question to be bumped to the home …
- 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
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
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
5
votes
Accepted
Solving a vector of quadratic equations
Shameless advertisement to a paper of mine: http://www.sciencedirect.com/science/article/pii/S0024379511004484 Quadratic vector equations, in Linear Algebra and its Applications, volume 438, 2013. Ar …
- 20.2k
11
votes
Accepted
name for a matrix operation
it is called "diagonal congruence" here. This makes sense, at least when $D$ is real, since it is a congruence. "Conjugate" sounds more like $D^{-1}AD$ or $\overline{A}$ to me.
- 20.2k
3
votes
Number of parameters needed to specify a Hermitian matrix of rank r.
Not sure if I am missing something here...
1) Rank-$r$ Hermitian matrices are determined uniquely by their image $U$ and how they act when restricted to $U$. The image can be any dimension-$r$ subspa …
- 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
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