# Tagged Questions

Questions about the properties of vector spaces and linear transformations, including linear systems in general.

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### Relation between eigenvalues of $A$ and $A^TA$?

For an $n\times n$ diagonizable matrix $A$, is there a relation between the eigenvalues of $A$ and the eigenvalues of $A^TA$? I ask this because I am looking into the relation between $A$ and $A+cI$, ...
180 views

### Complexity of approximating the size of the range of a matrix

Given an $m$ by $n$ matrix $M$ with $m \leq n$ and elements from $\{-1,1\}$, let us define: $$S_M = |\{Mx : x \in \{-1,1\}^n\}.$$ It is NP-hard to compute $S_M$ exactly I believe by applying the ...
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### Is there a nice choice-free argument to count the number of sublattices?

It's a well known fact that the number of index $n$ sublattices of a rank two lattice $\Lambda$ is given by $\sigma_1(n) = \sum_{d\mid n} d$. Here is a proof of this fact: Proof: choosing a basis of ...
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### How to prove this determinant is positive-II?

Question: Given an arbitrary number of real matrices of the form $A_i= \biggl(\begin{matrix} C_i+E_i & B_i \\ B_i^T & D_i-F_i \end{matrix} \biggr)$, where $B_i$ is an arbitrary $n\times n$ ...
Let $G$ be a digraph with adjacency matrix $A =(A_{ij})$ where $A_{ij}=1$ if and only if there is a directed edge $i \to j$ and $A_{ij}=0$ otherwise. Let $D= (D_{ij})$ be the degree matrix with ...