# Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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### Find all the $2 \times 2$ matrices $A$ satisfying $A^2 = 0$

Find all the $2 \times 2$ matrices $A$ satisfying $A^2 = 0$. So far I have this: But I don't know to proceed.
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### Is my understanding of matrices correct?

Let $V = R^2$ be our vector space with the unit base vectors $J(1, 0), K(0, 1)$. We have the linear map, $$T: V \to V$$ $$T(v) = v'$$ We can rewrite $\forall v \in V$ as a linear combination of the ...
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### Differentiating matrix exponential

I know that $$\frac{d}{dt}e^{At} = Ae^{At}$$ However, in one lecture, I found the following $$\frac{d}{dt}e^{A^Tt} = e^{A^Tt}A^T$$ The lecture is as follows How to show the second case? ...
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### Direct proof for angular velocity from direction cosine matrix

I am trying to work through the math of what should be a relatively simple proof of a direct definition of the angular velocity matrix starting from the direction cosine matrix. The reference for ...
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### Question regarding the similarity of an invertible matrix with its inverse .

Find the set $S$ of all possible $n×n$ invertible matrices $M$ such that $M$ is similar to $M^{-1}$ . My approach Actually, I was thinking about this problem when I came across a theorem stating ...
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### Finding n-th power of a 2*2 matrix with 2 identical eigen values

If$$A = \begin{pmatrix} 3 & -4 \\ 1 & -1 \\ \end{pmatrix}$$ prove that $$A^k = \begin{pmatrix} 1+2k & -4k \\ k & 1-2k \\ \end{pmatrix}$$ Now the first ...
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### Does the notion of a contraction operator depend on the norm which induces the metric?

Let $\lVert \cdot \rVert_a$ and $\lVert \cdot \rVert_b$ denote the $\ell_a$ and $\ell_b$ norms on $\mathbb{R}^n$ ($a,b$ are positive integers or $\infty$, $a \neq b$). Let $M_n(\mathbb{R})$ be the ...
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### How the Wronskian works

To prove linear independence of a set of functions, we say that given their Wronskian matrix $~W~$, $~W_x = 0~$ implies trivial solution $~(0,0,0,\cdots)~$ if the value (determinant) of the Wronskian ...
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### How to show the interpretability of NMF by a small qualitative example on a toy data?

In some paper, such as, Nonnegative Matrix Factorization: A Comprehensive Review, I see the interpretability of Nonnegative matrix factorization (NMF). However, I don't know the means of this. How to ...
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### How do I determine the transformation matrix T of the coordinate transformation from the base E to the base B?

In $\mathbb{R}^3$ the canonical basis $E=\left (\mathbf{e_1},\mathbf{e_2},\mathbf{e_3} \right )$ and $B=\left (\mathbf{b_1},\mathbf{b_2},\mathbf{b_3} \right )$ with $\mathbf{b_1}=(1,2,4)^T$, \$\...