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Results tagged with matrices
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user 290
Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level tag ra.rings-and-algebras, co.combinatorics, etc. and other tags that might be applicable. There are also several more specialized tags concerning matrices.
6
votes
Factorization of a matrix as a product of a symmetric and a skew-symmetric matrix
We have $M^T = B^T A^T = -BA$. Since $AB$ and $BA$ have the same characteristic polynomial, and so do $M$ and $M^T$, it follows that $M$ and $-M$ have the same characteristic polynomial. Equivalently, …
- 118k
5
votes
are intersections of kernels also kernels?
There is no such choice, if by "canonical" you mean natural in the category-theoretic sense. I am going to rename $V'$ to $W$.
You want to find a natural map $\text{Hom}(V \oplus V, W) \to \text{Hom} …
- 118k
10
votes
Accepted
Sum of the coefficients of the characteristic polynomial of periodic matrices
$\Box$
Q2: As before it suffices to consider block sums of companion matrices of cyclotomic polynomials. … this (the cyclotomic polynomials satisfy $\Phi_n(0) = 1$ for $n \ge 2$ so all these block matrices lie in $SL_n(\mathbb{Z})$ also). …
- 118k
13
votes
Existence of Rational Orthogonal Matrices
Consider matrices which fix $n-2$ of the standard basis vectors and describe a rotation in the plane spanned by the last two about an angle $\theta$ such that $\sin \theta, \cos \theta$ are both rational …
- 118k
6
votes
A truncated "geometric" matrix series
This is a long comment. Write $S$ for this sum. If we just directly imitate the usual geometric series argument we are led to consider
$$ASC = \sum_{k=1}^N A^k B C^k = S - B + A^N B C^N$$
so $ASC - S …
- 118k
6
votes
Formula for the entry of a matrix power
I agree with LSpice that I don't think this really needs a proof or a citation, but in combinatorics this sort of thing is often called "the transfer matrix method" and accordingly it is stated in com …
- 118k
5
votes
Accepted
Is it always possible to "separate" the eigenvalues of an integer matrix?
Think of $M$ first as a linear operator acting on $V = \mathbb{Q}^n$. Pass to a splitting field $K$ and consider the induced action on $V \otimes K$. This splits up into a direct sum of generalized ei …
- 118k
6
votes
Which positive definite symmetric matrices have solvable characteristic polynomial?
Here is a small observation. As a subspace of the space $\mathbb{Q}^n$ of monic polynomials of degree $n$ with rational coefficients, the solvable polynomials are dense (and so in particular are not c …
- 118k
5
votes
Accepted
Real matrix rings and associative hypercomplex numbers
It's not clear what you mean by "real matrix ring," which could either mean a ring of the form $M_n(\mathbb{R})$ or a real subalgebra of $M_n(\mathbb{R})$; if the latter, this is the same as saying "f …
- 118k
9
votes
Infinite matrices and the concept of "determinant"
First of all, "infinite matrices" aren't well-defined as linear transformations without additional hypotheses. … If you define the determinant of a matrix to be the product of its eigenvalues, then you run into immediate trouble: "infinite matrices" don't necessarily have any, even over an algebraically closed field …
- 118k
45
votes
Should the formula for the inverse of a 2x2 matrix be obvious?
Recall that the adjugate $\text{adj}(A)$ of a square matrix is a matrix that satisfies
$$A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = \det(A).$$
Like the determinant, the adjugate is multiplicati …
- 118k
18
votes
Are automorphisms of matrix algebras necessarily determinant preservers?
Here is a positive result. Every finite-dimensional algebra $A$ over a field $K$ has an intrinsic determinant, and in fact an intrinsic characteristic polynomial, which is preserved by all automorphis …
- 118k
17
votes
Accepted
A particular Lie algebra $L_{n}$ and (various) lie groups whose Lie algebra is isomorphic to...
The corresponding subgroup of $GL_n(\mathbb{R})$ consists of invertible matrices whose first column is $1, 0, 0, \dots$. This is the general affine group. …
- 118k
3
votes
Accepted
Heuristics for counting degrees of freedom
Let $V$ be a finite-dimensional real inner product space. You want to know the dimension of $\text{End}_G(V \otimes V^{\ast})$ where $G = O(V)$. Using the inner product we have an isomorphism $V \cong …
- 118k
12
votes
Expressing $-\operatorname{adj}(A)$ as a polynomial in $A$?
Lemma: The invertible $n \times n$ matrices are dense in the $n \times n$ matrices with the operator norm topology.
Proof. Let $A$ be a non-invertible $n \times n$ matrix, hence $\det A = 0$. …
- 118k