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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.
5
votes
Accepted
Finding the square root of a special matrix
Then the problem comes down to a question about Clifford algebras: Write $A = A^i\,x_i + R_2(x)$ where $R_2(x)$ vanishes to order $2$ in $x$ and the $A^i$ are $n$-by-$n$ matrices with entries in $\mathbb … only to to the relations $J^iJ^j+J^jJ^i = 2\delta^{ij} 1$, is known to have dimension $2^n$ and, since $n$ is even, it is also known to be isomorphic to $M_N(\mathbb{C})$, the algebra of $N$-by-$N$ matrices …
- 108k
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
7
votes
Accepted
Coordinate free isomorphism between $d+1$-dimensional antisymmetric rank $2$ tensors and $d$...
Here is a revised partial answer and some comments:
It seems that you are asking for some kind of isomorphism between $S^2(\mathbb{F}^d)$ and $\Lambda^2(\mathbb{F}^{d+1})$ that would 'have the greate …
- 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
$$
A^2 = \begin{pmatrix}126&5&95\\5&126&95\\95&95&150\ …
- 108k
15
votes
Accepted
$2 \times 2$ matrix question
Now consider the map $F$ from $\mathbb{R}^4$ into $\mathbb{R}^3$ (regarded as the traceless Hermitian $2$-by-$2$ matrices) defined by
$$
F(x) = \left[A +BV(x) + V(x)^*B^* + V(x)^*CV(x)\right]_0\,,
$$
where … For use below, define the norm on traceless Hermitian $2$-by-$2$-matrices $M$ by the rule $|M|^2 = \tfrac12 \mathrm{tr}(M^2)$. …
- 108k
3
votes
Accepted
Parametrization of real-valued SU(N)
In addition to the comments I made above about continuous solutions, I thought I'd point out a solution that works for all $n$ with only one point of discontinuity, namely
$$
(a_1\ a_2\ \ldots\ a_n) = …
- 108k
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
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
3
votes
standard form of antisymmetric matrix
Let $A_d$ denote the set of invertible $2d$-by-$2d$ skew-symmetric matrices, considered as an open subset of the vector space of all $2d$-by-$2d$ skew-symmetric matrices and let $\mathrm{GL}(2d,\mathbb … {R})$ denote the set of invertible $2d$-by-$2d$ matrices, considered as an open subset of the vector space of all $2d$-by-$2d$ matrices. …
- 108k
6
votes
Accepted
Is a pointwise " simple tensor" valued continuous map a tensor product of two continuous maps?
The answer is 'no'. For example, let
$$
X = \{\ B\otimes C\ | \det(B)\det(C) = 1\ \}\subset \mathrm{SL}(4,\mathbb{C}).
$$
Then $X$ is isomorphic to the group $\mathrm{SO}(4,\mathbb{C})$ and hence $\p …
- 108k
4
votes
Is it possible to obtain the vectors orthogonal to a given one by orthogonal transformations?
The kind of transformation that you want, i.e., a linear transformation $A$ such that $Ax\cdot x = 0$ for all $x$ and $|Ax|=|x|$, is possible only when the dimension of the space is even. The reason i …
- 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
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
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
6
votes
Accepted
Symplectic block-diagonalization of a complex symmetric matrix
The thing you want to think about is that the Lie algebra of the symplectic group is exactly the set of matrices of the form $JA$ where $A$ is symmetric, and you are trying to conjugate $JA$ into the Cartan … subalgebra consisting of the diagonal matrices of this form via the adjoint representation of the symplectic group on its Lie algebra. …
- 108k