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Results tagged with linear-algebra
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user 9025
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
1
vote
Solving $Ax=e_k$ for standard basis vector $e_k$, sparse $A$
Solve $A^T A \boldsymbol{x}=A^T\boldsymbol{e_k}$ first, it is small enough to handle by dense methods. If there is no solution, there is also none for the original. Otherwise get an (affine) basis o …
- 37k
2
votes
How many DISTINCT vectors we get from pairs v_i + v_j for some set of given vectors v_i ?
Your question is a little unclear, but I take it you want some sort of sampling method that estimates the number of distinct sums.
I'll answer a more general question. This method is not as well kno …
- 37k
2
votes
Accepted
Necessary conditions for existence of linear combination of these matrices to be singular
Eigenvalues are continuous functions of the matrix entries if this is expressed carefully. Consider $H(c) = cP_1+P_2+\cdots+P_m$. When $c$ is large and negative, the eigenvalues of $H(c)$ are all neg …
- 37k
1
vote
Fill the board with zeroes, inverting the intersections of rows and columns
This is an extended comment short of a solution.
Define an $n^2\times n^2$ matrix $A_n$ with rows and columns indexed by the positions $(i,j)$. Row $(i,j)$ of the matrix is 0 except for 1 in the col …
- 37k
2
votes
Accepted
When can an eigenvector be chosen uniquely which is invariant to permutation?
You can always do it if there is any eigenvector not summing to 0, and some other times too. First define $Aut(A)$ to be the group of permutation matrices $R$ such that $RAR^T=A$. If $A\boldsymbol{x …
- 37k
10
votes
The coefficient of a specific monomial in the expansion of the following polynomial
This is not an answer but a conjecture based on some trials.
$$a_{n,k} = \begin{cases} 1, & n=1 \\
0, & n\gt 1, k\text{ odd} \\
(-1)^{t\binom n2}\di …
- 37k
8
votes
Linear independence of element-wise powers of positive vectors
In the case of integer $\gamma_i$, the determinant is nonzero (in fact, strictly positive). This is proved in Gantmacher, The Theory of Matrices, Vol 2, p99. Gantmacher calls it a "generalized Vande …
- 37k
2
votes
interlacing roots/eigenvalues results and modern analogues
One fuzzy connection: Say matrix $A$ has characteristic polynomial $f(x)$. Then the derivative $f'(x)$ is the sum of the characteristic polynomials of the principal minors $A_{ii}$.
- 37k
16
votes
1
answer
2k
views
Overlapping Gershgorin disks
We all know Gershgorin's Circle Theorem, which I will summarise for convenience. Let $A=(a_{ij})$ be an $n\times n$ complex matrix. Define the disks $D_1,\ldots,D_n$ by $$D_i = \Bigl\{ z : |z-a_{ii}| …
- 37k
6
votes
2
answers
179
views
Eigenvalues of polynomials of two matrices
In this question, the matrices are square and real and the polynomials have real coefficients, but feel free to mention other fields if that is interesting.
Let $\chi(M)$ denote the characteristic pol …
- 37k
15
votes
Accepted
Why does an invertible complex symmetric matrix always have a complex symmetric square root?
Higham, in Functions of Matrices, Theorem 1.12, shows that the Jordan form definition is equivalent to a definition based on Hermite interpolation. That shows that the square root of a matrix $A$ (if …
- 37k
5
votes
Complex Eigenvalues of Directed Graphs
If $k$ is the greatest common divisor of the cycle lengths of the digraph, then the spectrum is invariant under rotation around the origin by $2\pi/k$. This is an application of the Perron-Frobenius t …
- 37k
2
votes
Computing the multiplicity of an eigenvalue of a 0-1 symmetric matrix...
The multiplicity of $\lambda$ is $n$ minus the rank of $\lambda I - A$, which you can find using Gaussian elimination. If you need to use floating-point arithmetic, you'll need to make decisions on w …
- 37k
1
vote
Accepted
Affine hull of a set of non-negative matrices with fixed row-sums
(The first part of this is the same as @Fedor's answer; I just carried out his algorithm.) Each row $i$ can be written as $a_i^{(1)}x_i^{(1)}+a_i^{(2)}x_i^{(2)}$ where $x_i^{(1)},x_i^{(2)}$ are non-ne …
- 37k
12
votes
How are eigenvalues and eigenvectors affected by adding the all-ones matrix?
A cute fact that is trivial to prove is this: define the characteristic polynomial of a matrix $M$ by $\phi_M(x) = |xI-M|$. Then for any $A$ and any $s$, $$\phi_{A+sJ}(x) = (1-s)\phi_A(x)+s\phi_{A+J}( …
- 37k