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Tagged with rank-1-matrices matrix-decomposition
6 questions
2
votes
2
answers
69
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Completing a rank-1 decomposition of a matrix
Let $M$ be an $m\times n$ matrix of rank $r$. I am interested to express $M$ as $C_1 + \ldots + C_r$ where each $C_i$ is a rank-1 matrix. How many $C_i$'s of rank-1 am I allowed to fix (if any) and ...
- 166
1
vote
1
answer
77
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Showing existence of symplectic transformations preserving a quadratic form
Question: I need help to prove the following statement.
Let $W_i:=w_iw_i^T\in\mathbb{R}^{n\times n}$, for $n$ even, be symmetric rank-1 matrices, $J=-J^T$ the canonical symplectic matrix and define ...
- 140
3
votes
1
answer
21k
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Finding the best rank-one approximation of the matrix $\bf A$
I have computed the singular value decomposition (SVD) of the following matrix $A$.
$$ {\bf A} := \begin{bmatrix}1&2\\0&1\\-1&0\\\end{bmatrix} = \underbrace{\left[\begin{matrix}0 & \...
- 1,400
1
vote
1
answer
2k
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All rank-1 matrices have an SVD
I have a rank-$1$ matrix $A \in \mathbb{R}^{m \times n}$ and a vector $u$ in its image. I could prove that the columns of $A$ are multiples of a vector $u$, and that $A$ can be written as $A = \alpha ...
- 13
1
vote
3
answers
591
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Factoring a given rank-$1$ matrix
Suppose you have a $n \times 1$ column vector
$$a=\begin{bmatrix}a_1\\{a_2}\\ \vdots\\{a_n}\end{bmatrix}$$
and a $1 \times m$ row vector
$$\quad b=\begin{bmatrix}b_1 & b_2 & \ldots & ...
- 131
4
votes
2
answers
544
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Recover vector $x$ from rank-$1$ matrix $Q=xx^H$
Let the matrix $Q \in\mathbb{C}^{n \times n}$ be known. It is also known that $Q=xx^H$, where $x=[x_1,\ldots,x_n]^T$ and $x^H$ is its conjugate transpose. What is $x$? How to recover it?
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