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Results tagged with matrices
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user 112641
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
Accepted
Block matrices and their determinants
The eigenvalues of $H$ are worked out in "The eigenvalues of some anti-tridiagonal Hankel matrices". …
- 1,593
6
votes
Accepted
Determinant involving traceless unitary hermitian matrices
Yes. It's real when $N\equiv 0 \text{ mod } 4$ and imaginary when $N\equiv 2\text{ mod } 4$.
The square of the determinant is $\det(A+iB)^2=\det(1-1+i(AB+BA))=i^N\det(AB+BA)$, so for either parity of …
- 1,593
4
votes
Accepted
How to find the analytical representation of eigenvalues of the matrix $G$?
It's straightforward to show that this is the inverse of $1/(N+1)$ times the tridiagonal matrix $T_N$ with $-2$ on its main diagonal and $1$ on its super- and sub-diagonals.
Let $t_N$ be the characte …
- 1,593
4
votes
Making binary matrix positive semidefinite by switching signs
The answer to part 1 is $\lceil \frac{n^2}{2}\rceil-n$. Any fewer than that and the vector $v$ of all $1$'s will satisfy $v^\top Av< \lfloor\frac{n^2}{2}\rfloor-\lceil \frac{n^2}{2}\rceil\le 0$. Let $ …
- 1,593
2
votes
Determinant of a block matrix with many $-1$'s
To complete Mahdi's answer, it suffices to show
$$(1-\sum_i\frac{x_i}{1+2x_i})\prod_i(1+2x_i)=\sum_{j\ge 0} (2-j)2^{j-1}e_j.$$
Clearly $\prod_i(1+ax_i)=\sum_{j\ge 0} a^je_j$, which explains the $2^je_ …
- 1,593