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Results tagged with matrices
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user 13972
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.
8
votes
Accepted
Diagonalization of symmetric matrices of functions
.
$$
Now, consider the map $s:\mathrm{SL}(2,\mathbb{R})\to H$ (where $H$, a hyperboloid of one sheet, is the quadric surface in the symmetric $2$-by-$2$ matrices defined by setting the determinant equal …
- 108k
6
votes
Accepted
Analytical form for the nuclear norm of an $n \times n$ matrix
As Gro-Tsen pointed out, I had not computed the Galois group of the governing polynomial, and, in fact, my original answer was wrong. I believe that the following answer is correct, though.
If by 'li …
- 108k
3
votes
Does linearity of cofactor imply linearity of determinant for 3×3 symmetric matrices?
Here's a counterexample to the OP's literal question: Consider the following four symmetric (in fact, diagonal) $3$-by-$3$ matrices: $A_1 = \mathrm{diag}(0,0,0)$, $A_2 = \mathrm{diag}(\frac12,3,3)$, … To see this, note first that the cofactor function on $3$-by-$3$ matrices $A\mapsto \mathrm{cof}(A)$ is a quadratic function (i.e., the mapping $\beta(A,B) = \mathrm{cof}(A+B) - \mathrm{cof}(A)-\mathrm …
- 108k
8
votes
Linear subspaces of $\mathrm{GL}_n(\mathbb{R})$ whose inverses are also linear subspaces
Second, the subspace $\Sigma_n\subset M_n(\mathbb{R})$ consisting of the symmetric $n$-by-$n$ matrices has the property that its set of invertible elements is linear in the OP's sense. … For example, when $n=4$, take $S'\subset M_n(\mathbb{R})$ to be the $3$-dimensional subspace of matrices of the form
$$
s = \begin{pmatrix} a & b & c & 0\\-b&a&0&-c\\-c&0&a&b\\0&c&-b&a\end{pmatrix} = a …
- 12.9k
11
votes
Accepted
Maximal compact subgroup of $\mathrm{SL}(2,\mathbb{H})$
Consider the mapping $\sigma:\GL(2,\mathbb{H})\to M_2(\mathbb{H})$ given by
$$
\sigma(A) = A^* A
$$
where $A^*$ is the conjugate transpose of $A$ in $M_2(\mathbb{H})$, the $2$-by-$2$ matrices with entries … Consequently, the image of $\sigma$ lies in the $6$-dimensional real subspace $S_2(\mathbb{H})$, consisting of the matrices $s\in M_2(\mathbb{H})$ that satisfy $s = s^*$. …
- 12.9k
11
votes
Diagonalizing quaternionic unitary matrices
This follows from the general fact that, in a compact connected Lie group, every element is conjugate to an element in a maximal torus (and all maximal tori are conjugate). This result is proved in j …
- 108k
26
votes
Accepted
Square root of doubly positive symmetric matrices
No. If $$A = \begin{pmatrix}10&-1&5\\-1&10&5\\5&5&10\end{pmatrix},$$ then $A$ is positive definite but does not have all entries positive, while
$$
A^2 = \begin{pmatrix}126&5&95\\5&126&95\\95&95&150\ …
- 108k
10
votes
Accepted
Characteristic polynomial of an $8 \times 8$ symmetric matrix with indeterminate entries rel...
, it follows that $L$ is conjugate in $\mathrm{SO}(8)$ to an element of the maximal torus ${\mathrm{SO}(2)}^4$, i.e., a blocked diagonal matrix
where the diagonal elements are the $2$-by-$2$ rotation matrices …
- 108k
6
votes
Subgroup of $\mathrm{GL}_n$ stabilizing linear subspace skew-symmetric matrices
Here is an outline of the argument that shows that the $\mathrm{SL}_6(\mathbb{C})$-stabilizer of the generic $3$-plane $W\subset\Lambda^2(\mathbb{C}^6)$ has dimension $1$, not $0$, as (apparently) cla …
- 108k
3
votes
Accepted
Parametrization of real-valued SU(N)
In addition to the comments I made above about continuous solutions, I thought I'd point out a solution that works for all $n$ with only one point of discontinuity, namely
$$
(a_1\ a_2\ \ldots\ a_n) = …
- 108k
3
votes
Accepted
Problem arising in metrizability of connections: Simultaneously skewsymmetrizing matrices
The reason is the following: Let $W$ be the span of the matrices $S_1,\ldots,S_m$. … and are such that any two unimodular matrices $U$ and $V$ that conjugate the $S_i$ into skew-symmetric matrices must differ by an orthogonal matrix, then the answer (for $n>2$) is $m=2$. …
- 108k
28
votes
Is the linear span of special orthogonal matrices equal to the whole space of $N\times N$ ma...
For $n>2$, the span of the matrices in $\mathrm{SO}(n)$ is the full space $M_n(\mathbb{R})$ of $n$-by-$n$ matrices with real entries. …
- 108k
23
votes
Accepted
Existence of double eigenvalue
The generic pair $A$ and $B$ of $4$-by-$4$ Hermitian symmetric matrices will not have any nonzero real linear combination that has a double eigenvalue. … Added Remark: To see the claim that this property holds for a generic linearly independent pair of Hermitian symmetric $4$-by-$4$ matrices $A$ and $B$, it is only necessary to observe the following: …
- 108k
15
votes
Accepted
$2 \times 2$ matrix question
Now consider the map $F$ from $\mathbb{R}^4$ into $\mathbb{R}^3$ (regarded as the traceless Hermitian $2$-by-$2$ matrices) defined by
$$
F(x) = \left[A +BV(x) + V(x)^*B^* + V(x)^*CV(x)\right]_0\,,
$$
where … For use below, define the norm on traceless Hermitian $2$-by-$2$-matrices $M$ by the rule $|M|^2 = \tfrac12 \mathrm{tr}(M^2)$. …
- 108k
3
votes
Regularity for the roots of (characteristic) polynomials with given multiplicity
I think that there is a smooth (or analytic) result of the kind that you are seeking:
Let $M^m$ be a connected smooth (or analytic) manifold, and let $P:M\times\mathbb{R}\to\mathbb{R}$ be a smooth …
- 108k