# Questions tagged [linear-algebra]

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

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### When does inverse of a real matrix have positive diagonal entries?

This question originates from a numerical simulation where we observe that all diagonal entries of inverses of an ensemble of real matrices are positive. We expect there should be a reason for this ...

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### Do we know what the impulse to “introduce” the Jordan canonical form was?

Mo-ers,
Do you know how it was that the study of the Jordan canonical form began?
There are certain things that may be said once one has thought about the matter: for instance, one can say that the ...

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58 views

### Vectors linear dependence/ independence [on hold]

Let A be k x k matrix with real entries and x ≠ 0. Then the vectors x, Ax, A2x, A3x, A4x, A5x …….AK x are
linear dependence/ independence cannot be determined from given data
linearly independent
...

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47 views

### How can I find the integral orthogonal group of a given symmetric positive definite form?

I wonder how one can study the integral orthogonal group of a given (symmetric, positive definite) bilinear form like the one described by the following matrix:
$$M=\begin{bmatrix}
x_1 &...

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59 views

### How to calculate the volume of a section of a convex body?

The following is essentially a partial case for my previous question.
Let $B\subset\mathbb{R}^m$ be the unit ball with respect to a concrete norm on $\mathbb{R}^m$, say $l^p$-norm, $p\in (1,\infty)$....

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47 views

### How do I test two square matrices are transpose to each other if only the column vector summations are known?

Given two secret square matrices, say $\left( {\begin{array}{*{20}{c}}
{{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\
{{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\
{{a_{31}}}&{{a_{32}}}&{{a_{33}}}
\...

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378 views

### Zero tensor product over a complex algebra?

Let $A$ be an algebra over $\mathbb{C}$. Let $M$ be a left $A$-module, let $N$ be a right $A$-module and consider the tensor product $N \otimes_A M$, which is a complex vector space.
Q1: Can this ...

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44 views

### Uniqueness of projection under spectral norm

I am considering
$$
\min_{M\in \mathcal{M}} \|X - M\|:=x \neq 0,
$$
where $X$, $M$ are $m\times n$ matrices, $\|\cdot\|$ is spectral norm and $\mathcal{M}$ is a matrix subspace. I wonder to what ...

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64 views

### Algebra of block matrices with scalar diagonals

I am interested in block matrices $A$, that is $A\in M_{n\times n}(R)$ where $R=M_{s\times s}(k)$ and $k$ is a field, such that for every positive integer $m$ the matrix $A^m$ has only scalar blocks ...

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**1**answer

75 views

### Lipschitz property of matrix function only depending on singular values

Let $f$ be a function from $\mathbb{R}^{n\times n}$ to $\mathbb{R}$ such that there exists another symmetric function $g$ (invariant under permutation of coordinates) from $\mathbb{R}^{n}$ to $\mathbb{...

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188 views

### Interpretation of determinants on commutative rings

In real Euclidian space, the result of the determinant can be interpreted as the oriented volume of the image of the unit cube under an invertible linear map.
This interpretation conceptually depends ...

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votes

**1**answer

215 views

### Is there a general geometric characterization for polynomials to be linearly dependent?

Consider $P$ the complex projective plane, and fix a line $L$ in $P$
I had a conjecture, that prof. I. Dolgachev showed me how to prove, that $3$ quadratic polynomials depending on a variable $z \in ...

**5**

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109 views

### How to calculate the volume of a parallelepiped in a normed space?

Let $E$ be a real normed space, and let $v_1,...,v_n\in E$ be linearly independent. The parallelepiped defined by these vectors is $P=\{\sum_{i=1}^{n}\alpha_i v_i|~0\le\alpha_i\le 1\}$. Since $E$ is a ...

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### Is there an easy characterisation (perhaps some generalised Löwner representation) for operator monotone functions of order $n$?

As per my understanding, roughly stated, $f$ is an operator monotone function of order $n$ if for all $n\times n$ (Hermitian) matrices, $X,Y\ge0$ which satisfy $X\ge Y$, we have $f(X)\ge f(Y)$.
If $f$...

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**1**answer

108 views

### Positive definite matrix

We have $a_1,a_2,...,a_n\in (0,1)$ and matrix
M=
\begin{bmatrix}2a_1&a_2&a_3&.&.\\a_2&2a_2&a_3&.&.\\a_3&a_3&2a_3&.&.\\.&.&.&.&.\end{...