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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.
30
votes
Accepted
Matrix equation $XAXBXC=I$
Here is an argument showing that the answer is 'yes'. I'll let you check the details and that this result generalizes to all higher degrees.
Consider the map $f_{ABC}:\mathrm{U}(n)\to\mathrm{U}(n) …
- 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
27
votes
Accepted
Alternate and symmetric matrices
I feel that framing this question in terms of matrices rather than bilinear forms on a vector space obscures what is actually going on and makes it harder to understand what needs to be proved. …
- 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
$$
A^2 = \begin{pmatrix}126&5&95\\5&126&95\\95&95&150\ …
- 108k
23
votes
Accepted
Existence of double eigenvalue
The generic pair $A$ and $B$ of $4$-by-$4$ Hermitian symmetric matrices will not have any nonzero real linear combination that has a double eigenvalue. … Added Remark: To see the claim that this property holds for a generic linearly independent pair of Hermitian symmetric $4$-by-$4$ matrices $A$ and $B$, it is only necessary to observe the following: …
- 108k
15
votes
Accepted
$2 \times 2$ matrix question
Now consider the map $F$ from $\mathbb{R}^4$ into $\mathbb{R}^3$ (regarded as the traceless Hermitian $2$-by-$2$ matrices) defined by
$$
F(x) = \left[A +BV(x) + V(x)^*B^* + V(x)^*CV(x)\right]_0\,,
$$
where … For use below, define the norm on traceless Hermitian $2$-by-$2$-matrices $M$ by the rule $|M|^2 = \tfrac12 \mathrm{tr}(M^2)$. …
- 108k
15
votes
Accepted
What it is the volume of the unit ball section of the cone of positive definite matrices?
Using the operator norm, as you have defined it, the fraction of the unit ball in real symmetric $n$-by-$n$ matrices that consists of positive definite matrices is $2^{-n(n+1)/2}$. … 0.2493898621\ .
$$
Thus, nearly one-quarter of the unit ball in this case consists of positive definite matrices. …
- 108k
15
votes
Parametrization of positive semidefinite matrices
To get a parameterization of the kind you want, the space $S_{n,r}$ of positive semidefinite symmetric $n$-by-$n$ matrices of rank $r$ (with ($0<r<n$) would have to be contractible, but it is not. … For example, in the first nontrivial case $r=1$, this is the space $S_{n,1}$ of rank $1$ symmetric $n$-by-$n$ matrices, and every $A\in S_{n,1}$ can be written in the form $A = vv^T$ where $v$ is a nonzero …
- 108k
12
votes
Accepted
Is there any connection between this matrices
I assume that the problem is to try to determine which pairs $(P_1,P_2)$ of positive definite Hermitian symmetric $N$-by-$N$ matrices can be written in the above form for some pair $(H_1,H_2)$ of positive … semi-definite Hermitian symmetric $N$-by-$N$ matrices. …
- 108k
12
votes
Accepted
Explicit Isomorphism between $Cl(8)$ and $\mathbb{R}(16)$
Here's a standard explicit formula: Let $\mathbb{O}\simeq\mathbb{R}^8$ denote the algebra of octonions, and for $x\in\mathbb{O}$, let $L_x$ (respectively $R_x)$ denote the linear map from $\mathbb{O} …
- 108k
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
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 $s = s^*$. …
- 108k
10
votes
The probability for a symmetric matrix to be positive definite
If you use the fact that the group $\mathrm{SO}(n)$ acts on the symmetric matrices in the obvious way and parametrize the 'hemisphere' of symmetric matrices of positive trace and norm $1$ by the symmetric … matrices of zero trace via the $\mathrm{SO}(n)$-equivariant mapping
$$
P = \frac{(I+z)}{(1+|z|^2)^{1/2}},
$$
then you can cover the symmetric matrices of trace zero by acting by $\mathrm{SO}(n)$ on the …
- 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
10
votes
Accepted
$SO(N^2-1)$ and the adjoint representation of $SU(N)$
Actually, it does not look like that. Take the case $N=3$. The representation of $\mathrm{SU}(3)$ on ${\frak{so}}(8)$ breaks up into the $8$-dimensional subspace ${\frak{su}}(3)$ and an irreducible …
- 108k