Questions tagged [linear-algebra]

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 concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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Range of $f(x)=[x^2 - 3x]$ [on hold]

I need to find the range of this function $$f(x)=gif (x^2 - 3x.)$$ I have no idea how to start solving this. Can anyone help?
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Simplex optimization: how to project to nearest feasible point, with box constraints and total constraint

I am working in a simplex-based constrained optimization setting, wherein several texts suggest that algorithms such as Nelder-Mead can be modified for constraints by projecting an infeasible point ...
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Defective Matrices Transforms

Defective Matrix:$\begin{bmatrix}1&1\\0&1\end{bmatrix}$ I am unable to wrap my head around what kind of transforms yield defective matrices or matrices where geometric multiplicity is less ...
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Strong separation of symmetric rank $1$ matrices

Let $V$ be the inner product space of $n \times n$ symmetric matrices where the inner product is given by $\langle A, B \rangle = trace(AB)$ and let $A, B \in V$ be such that they have eigenvalue $1$ ...
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Same Eigenvalues = Similar Matrices?

While watching Strang's lecture on similar matrices he stated that if any two matrices,$A$ and $B$, have the same eigenvalues then they can be put in the form $A=P B P^{-1}$ . This is very easy to see ...
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If $\operatorname{Tr}(H)\ge 0$, is $\operatorname{Tr}(PH)\ge 0?\$ $H$ is Hermitian, $P$ is positive definite.

Old: Given $\operatorname{Tr}(DH)\ge 0$ for positive real diagonal $D$ and Hermitian $H$, is $\operatorname{Tr}(H)\ge 0$? Let $D$ be positive real diagonal and $H$ be Hermitian such that $H$ may have ...
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Polynomial in a $2$-dimensional Euclidean space

I am reading about the problem of moments and get some troubles in they way one defines the measure on some $k$-dimensional Euclidean space by polynomial. Could you please help me? More precisely, I ...
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Whether there is always an intersection between Column Space and Nullspace [on hold]

Suppose there is a matrix $A$ that satisfies $Ax=b$, and it have two solutions $c$ and $d$. Then we can know $Ac=b$ and $Ad=b$, so $A(c-d)=0$. But $c$ and $d$ are the two column vectors of matrix $A$, ...
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Is set of matrices form $M_{3\times 3}(\mathbb R)$ with rank 1 is closed or open?

Is set of matrices form $M_{3\times 3}(\mathbb R)$ with rank 1 is closed or open ? I know that $M_{3\times 3}(\mathbb R)$ can be viewed as $\mathbb R^9$ with the euclidean norm on it. But I do not ...
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Can a polynomial have repeated complex roots?

Can a polynomial have repeated complex roots - or is it only possible that it can have repeated real roots? If so, can you please also provide an example of a polynomial with complex roots where you ...
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Can you factor equations which has $e^{-x}$ and $x$ together?

I am working on a math investigation, and I got a function: $y = 5x - 2e^{-4x} + 2$. The problem is, I want to change this equation such that $x$ is in terms of $y$. Is there a way so? Or is it ...
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Proof that a certain set of polynomials is linearly independent [on hold]

For $c_0, \ldots, c_n$ pairwise distinct complex numbers. Show that the polynomials $((X-c_i)^n)_{0 \leq i \leq n}$ are linearly independent. I need to prove it without induction. Any help would be ...
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Eigenvalues for fingerprint classification

i acquired waveshare's (uart capacitive) fingerprint scanner which provides me with eigenvalues of a finger. these values are used to compare/classify fingerprints as per the scanners documentation. ...
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Orientation for a vector spaces determines a canonical orientation for the dual vector space

Suppose $V$ is an $n$-dimensional real vector space, with $n > 0$. Show that an orientation for $V$ determines a canonical orientation for $V^*$, the dual of $V$. The idea I had in mind to show ...
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Clarification on the meaning of “uniqueness” in the (linked) Matrix diagonalization theorem

I came across this source yesterday, where they say: given a square real matrix S of size $nxn$, with n linearly independent eigenvectors, there exists an eigendecomposition such as $S=UDU^{-1}$. If ...
The unitarity of a given operator/matrix $A$ depends on the underlying inner product, so I guess it in principle possible for a given non-unitary $A$ to act as a unitary with respect to a different ...