# Questions tagged [rank-1-matrices]

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### Characteristic polynomial of convex combination

Given the following lemma on rank-$1$ matrices which I think I understand How could I deduce the following on the characteristic polynomial of $A_s$? I don't get it just using multilinearity of ...
• 5,514
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### Let A be a complex $n \times n$ matrix with rank(A) = 1. Why is the minimal polynomial $x(x-Tr(A))$? [duplicate]

We know $rank(A) = 1$ so I have $n-1$ eigenvalues which are $0$ So my characteristic polynomial is $(x-0)^{n-1}(x-a) = x^{n-1}(x-a)$ with $a$ the last eigenvalue to determine. Now, I found on some ...
1 vote
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• 539
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### Explaining the non convexity of rank 1 matrix

I understand from this post: How can we show/prove that a rank-$1$ matrix is non-convex? that a rank-1 matrix is non convex but I don't understand the proof. I know that the summation of two convex ...
92 views

### Is this rank-$1$ (complex) matrix positive semidefinite?

In Is this rank-$1$ matrix semidefinite?, I have seen that $X = xx^T$ is PSD when $x$ is real. What about the case when $X$ is Hermitian? I know that it is PSD but I'm not exactly sure how to prove it....
103 views

### What does "rank one" in "rank one constraint system" mean?

Although rank one constraint system (r1cs) is heavily used in zero-knowledge proofs, I didn't find anywhere details on where the rank comes from.
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### Is the multiplication of rank one matrices always yield rank one matrix? [closed]

Let's say we have infinite multiplication of rank one matrices. Does this yield rank one matrix?
1 vote
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### Can this $3 \times 3$ tridiagonal Toeplitz matrix be rank-$1$?

I am trying to determine whether the following tridiagonal $3 \times 3$ matrix can have a rank of $1$. $$\begin{bmatrix}a&b&0\\b&a&b\\0&b&a\end{bmatrix}$$ For $a = b = 0$, the ...
365 views

### $3 \times3$ matrices with rank $1$ or $2$ are manifolds

Let $M, N \subset \mathbb{R}^{n}$ be sets of matrices $3 \times 3$ with rank $1$ and $2$, respectively. Show that $M$ and $N$ are manifolds such that dim $M=5$ and dim $N=8$. I have thought about ...
193 views

### Find out the convex hull of the set $\left\{\pm \mathbf{u} \mathbf{u}^{T} \mid\|\mathbf{u}\|=1\right\}$ in a compact form ($u$ is a n-d vector)

According to the answer from @Cloudscape The first step of finding the convex hull of a given set would be to visualize the convex hull and guess it. The second step would be to prove your guess ...
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### $2$-norm of a rank-$1$ matrix

I want to prove that $\|A\|_2 = \|x\|_2\|y\|_2$ given that $A = xy^T$ is a rank one matrix. This is my incomplete attempt so far, I get stuck when I need to take into account the spectral radius of ...
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1 vote
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### How can I solve "average" best rank-$1$ approximation?

Assume I want to minimise this $$\min_{x,y} \left\| A - x y^T \right\|_{\text{F}}^2$$ then I am finding best rank-$1$ approximation of $A$ in the squared-error sense and this can be done via the SVD, ...
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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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1 vote
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### Element-wise upper bound by rank-1 matrix

I would like to solve the following optimization problem for vectors $\mathbf{u} \in \mathbb{R}^m$ and $\mathbf{v} \in \mathbb{R}^n$, given a matrix $\mathbf{h} \in \mathbb{R}_{\geq 0}^{m \times n}$ ...
• 158
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### Applications where rank-1 matrices are useful

I am trying to list down applications where having a rank-1 matrix is advantageous. I know only of 2D convolution which boils down to a series of 1D convolutions if filter response is separable. Can ...
• 987
1 vote
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### Is the product of a diagonal matrix and a rank-$1$ matrix still rank-$1$?

First let me make two statements to give my question the proper context. Consider $D$ a diagonal matrix and $u v^T$ a rank 1 matrix. From my current knowledge there exist computationally cheap ...
• 97
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### Some basic questions regarding rank-$1$ matrices

If an $n \times n$ matrix $B$ has rank $1$, and $A$ is another $n \times n$ matrix, then why does $A B$ also have rank $1$? This showed up in a solution that I read through, but it doesn't seem like ...
75k views

### A rank-one matrix is the product of two vectors

Let $A$ be an $n\times m$ matrix. Prove that $\operatorname{rank} (A) = 1$ if and only if there exist column vectors $v \in \mathbb{R}^n$ and $w \in \mathbb{R}^m$ such that $A=vw^t$. Progress: I'm ...
2k views

### Rank-$1$ matrices

I am struggling to understand the lecture notes (I have not learned linear algebra, but I do know some basic stuff). I understand the concept of rank of a matrix. A rank of matrix $A$ is given by the ...
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1 vote
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### How to get orthogonal rank-$1$ approximations?

Given $k$ matrices $A_1, A_2, \dots, A_k \in {\Bbb R}^{m \times n}$, I would like to find the matrices $\tilde{A}_1, \tilde{A}_2, \dots, \tilde{A}_k$ such that $\tilde{A}_i$ is a rank-$1$ ...
839 views

### Given matrices $A$ and $B$, how can I find a scalar $s$ that makes $A + s B$ rank-$1$?

Given $3 \times 3$ matrices $A$ and $B$, how can I find a scalar $s$ that makes the matrix $A + s B$ rank-$1$? Is there a method using singular value decomposition or eigenvalues? Thanks!
444 views

### Singularity of a conic combination of rank-$1$ matrices

Given rank-$1$ square matrices $A_1, A_2, \dots, A_n$, determine if there exists $x \in \mathbb R_{>0}^n$ such that $$\sum_{i=1}^n x_i A_i = x_1 A_1 + \cdots + x_n A_n$$ is singular, or decide ...
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### Generation of rank-$2$ matrices from a dictionary of rank-$1$ matrices
Consider the set $$\mathcal{D} = \left\{ \mathbf{A} \in \mathbb{C}^{2 \times 2} \mid \mbox{rank} (\mathbf{A}) = 1, \|\mathbf{A}\|_{F} = 1 \right\}$$ of all $2 \times2$ rank-$1$ matrices with unit norm ...