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Results tagged with matrices
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user 38468
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.
3
votes
Accepted
Proving 2 matrices have the same trace
Let's call $C=A-B$. Then you have $C^2=I_n$ and $BC-CB$ is invertible. You want to show that $C$ has an equal dimension of $1$ and $(-1)$ eigenspaces, which in turn implies both the equality $tr(A)=tr …
- 5,186
10
votes
Accepted
Eigenvalues of a matrix with entries involving combinatorics
One can prove this statement along the following lines.
Prove that ${\rm Trace}(M(l,n)) = 1+l +\cdots + l^{n-1}$.
Prove that $M(l,n)^p = M(l^p,n)$.
Clearly, these statements 1 and 2 together imply …
- 5,186
26
votes
1
answer
1k
views
Real square roots of symmetric matrices
In joint work with Andreas Fischle (TU Dresden, Germany) and Patrizio Neff (U Essen, Germany) we needed to use the following statement: If $S$ is a real $n\times n$ matrix with $S^2$ symmetric, then t …
- 5,186
7
votes
Accepted
Proof for a rank-one decomposition theorem of positive (semi) definite matrices
I don't know what's going on with the paper, but here is an argument regarding existence of such decompositions.
Given a rank one decomposition
$$X = \sum_{i=1}^R x_ix_i^T$$
one has $\sum_{i=1}^R x_ …
- 5,186
15
votes
Accepted
Die hard nilpotent spaces
Consider the subspace of $9 \times 9$ uppertriangular matrices spanned by
$$
A=\left(
\begin{array}{ccc}P&0&0\\0&P&0\\0&0&0\end{array}
\right),~~
B=
\left(
\begin{array}{cc}0&0&0\\0&P&0\\0&0&P\end{array …
- 5,186
2
votes
When is this matrix singular?
For $\phi_j=0$ and $t_k=k$ you have (up to a constant) the matrix $A_{jk}= \lambda_j^k - \lambda_j^{-k}$ where $\lambda_j=\exp( i w_j)$. You can factor out $\lambda_j-\lambda_j^{-1}$
to get a matrix $ …
- 5,186
11
votes
2
answers
818
views
Determinant and eigenvalues of a specific matrix
This came up in a conversation with an engineer friend of mine.
Let $c>0$ be a constant. Let $A_{ij}$ be an $n$ by $n$ matrix with entries
$$
A_{ij} = e^{-c(i-j)^2}.
$$
Is there a name for this matri …
- 5,186
2
votes
Accepted
For a set of matrices $S$, find $X$ such that the elements of $SX$ commute
Also, even if $A_i$ are nice, say sparse, the matrices $A_iA_j^{-1}$ may not be sparse at all. Still, it's something.
It might be easier if one of $A_iA_j^{-1}$ has distinct eigenvalues. …
- 5,186