# Tagged Questions

Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

3k views

925 views

532 views

### Generalized Eigenvalue Problem with one matrix having low rank

I have a specific Generalized Eigenvalue Problem (GEVP) where i am primary not interested in solving this problem but concluding from a standard EVP the spectrum of the GEVP. The Problem Let $A$ be ...
81 views

### Algebra defined by $a^2=a,b^2=b,c^2=c,(a+b+c)^2=a+b+c$

Let $\cal A$ be the (noncommutative) unitary $\mathbb Z$-algebra defined by three generators $a,b,c$ and four relations $a^2=a,b^2=b,c^2=c,(a+b+c)^2=a+b+c$. Is it true that $ab\neq 0$ in $A$ ? This ...
84 views

### Does anyone know a reference to best-fitting lines with integral coefficients?

I'm writing up a manual on how to generate "nice" Linear Algebra problems; that is, where the solutions tend to be integral. I "discovered" the following fact about the best-fitting line: Theorem. ...
79 views

### In a finite dimensional inner product space with $T ∈ L(V)$, show that $\langle u,v\rangle = \langle T(u),T(v)\rangle$ implies $T$ is invertible.

Here is how I've tried to go about it, and I'm curious if it's true or if I'm way off base. T is invertible iff null$(T)=\{0\}$. Let $v∈V$ and suppose $T(v)=0$. If we can show that $v=0$, then $T$ is ...
78 views

106 views

137 views

### Linear Algebra Book Recommendation like Tao's Analysis

I am looking for a book that explains Linear Algebra, where it is build from axioms to higher level of Linear Algebra. It does not have to be a book on elementary level. As example from other fields, ...
75 views

### Solving a matrix differential equation

I am trying to solve: $\frac{d U_t}{dt} = Tr(G^{\dagger}U_t)G - Tr(U_t^{\dagger}G)U_t G^{\dagger} U_t$ Where $U_t \in SU(4)$ and $G \in SU(4)$ is given and constant. Is it possible to solve this ...
196 views

### Hadamard matrices and sub-matrices (Converse of Sylvester Construction)

Let $H$ be a $d$ by $d$ real Hadamard matrix, namely: $$HH^{T}=d I$$ where $I$ is the identity matrix and $d=2^{k}$ for some natural number $k\geq 2$. The entries of $H$ are either $1$ or $-1$ and it ...
164 views

### Exactly $n-1$ nonzero elements if $\det(A)=0$ for every arrangement

Let $x_1,x_2,\dots,x_{n^2}\in\mathbb{R}$ with the property that any $n\times n$ matrix with exactly these elements has determinant $0$. Suppose also that there are at least $n$ distinct elements. ...
242 views

### Proof of the conjecture that the kernel is of dimension 2, extended

Pursuing my research, I am now looking for a proof of an extension of the problem proposed here and answered. It's an extension in the sense that I'm now considering two different $t_1$ and $t_2$. The ...
Let $G$ denote the groups of $n\times n$ invertible matrices and $H$ be the subgroup of invertible upper triangular matrices. For $n=2$, by row reduction, or equivalently LU decomposition, it is ...