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# Questions tagged [linear-algebra]

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

3,769 questions
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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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### 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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### 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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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 $\mathbb{... 0answers 188 views ### Interpretation of determinants on commutative rings In real Euclidian space, the result of the determinant can be interpreted as the oriented volume of the image of the unit cube under an invertible linear map. This interpretation conceptually depends ... 1answer 215 views ### Is there a general geometric characterization for polynomials to be linearly dependent? Consider$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 ...
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### 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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### 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{...

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