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Results tagged with matrix-theory
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user 4312
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.
14
votes
Accepted
Rank of $A\otimes B - B\otimes A$
For generic $A,B$ the matrix $A$ is invertible and $B=CA$, where $C$ is also generic. We have $(A\otimes B-B\otimes A)(u\otimes v)=Au\otimes CAv-CAu\otimes Av$. The vectors $Au$ run over the whole $\m …
- 109k
19
votes
When the sum of positive definite matrices converges, does the sum of the norm of the associ...
Yes. The norm of a positive definite matrix does not exceed its trace, and the sum of traces is finite, since the sum of diagonal elements is finite for each of $n$ places.
- 109k
7
votes
Accepted
Rank of a combinatorial matrix
If $(c_0,\dots,c_{2n-1})$ is a row of $A$, we look at a polynomial $c_0-c_1s+\dots-c_{2n-1}s^{2n-1}$. We have to find the dimension of the subspace $\Sigma$ formed by such polynomials in the space $\P …
- 109k
1
vote
Accepted
Existence of a matrix in $\mathbb{F}_2$ with some invertibility properties
There is no such $n$. Assume that it exists, then $n=6$ also satisfies this property. Denote the rows of $X$ by $r_1,\ldots,r_6$, the columns of $X$ by $c_1,\ldots,c_{10}$, the rows of $I$ by $e_1,\ld …
- 109k
6
votes
Product of a Finite Number of Matrices Related to Roots of Unity
Lemma 1. Let $\eta$ denote primitive $m$-th root of 1. Consider all
$m$-tuples $0\leq a_1<a_2<\dots< a_m\leq 2m$ such that
either $a_1>0$ or $a_{m}<2m$. Take the sum of $\eta^{a_1+\dots+a_m}$
over all …
- 109k
3
votes
Accepted
Condition for non-vanishing trace
Your trace equals ${\rm tr} ((BPA^\top+APB^\top)X)$. This equals 0 for all symmetric matrices $X$ if and only if $C=BPA^\top+APB^\top=0$ (else take $X=C$, note that $C$ is symmetric). Of course, this …
- 109k
10
votes
Accepted
The determinant of a $4\times4$ matrix associated to some specific polynomial as follow
For a given monomial $Y=\frac{x_{i_1}\cdots x_{i_k}}{x_{j_1}\cdots x_{j_k}}$ the coefficient $L(Y)$ multiplied by the constant $(-1)^{\sum_{i<j} a_{ij}}$ equals $$[Y]\prod_{i,j}(1-x_i/x_j)^{a_{ij}}=\i …
- 109k
5
votes
How to prove that each element of $A(A^TA)^{-1}A^Ty$ is greater than 0, if $A(i,j)>0$ and $y...
This does not hold in general.
Note that $\beta=A(A^TA)^{-1}A^Ty\in \mathbb{R}^m$ belongs to the image ${\rm Im} A$ (that is, to the column space of $A$). Also, $A^T\beta=A^TA(A^TA)^{-1}A^Ty=A^Ty$, so …
- 109k