Mathematics

Questions tagged [rank-1-matrices]

85 questions
83
votes
3 answers

Eigenvalues of the rank one matrix $uv^T$

Suppose $A=uv^T$ where $u$ and $v$ are non-zero column vectors in ${\mathbb R}^n$, $n\geq 3$. $\lambda=0$ is an eigenvalue of $A$ since $A$ is not of full rank. $\lambda=v^Tu$ is also an eigenvalue of $A$ since $$Au = (uv^T)u=u(v^Tu)=(v^Tu)u.$$ Here…
user9464
49
votes
3 answers

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 going back and forth between using the definitions…
coconutbandit
  • 493
26
votes
6 answers

Rank of sum of rank-$1$ matrices

If you sum a certain number of rank- $1$ matrices: $$X = u_1 u_1^T + u_2 u_2^T + \cdots + u_N u_N^T$$ Is the result guaranteed to be rank- $N$ assuming the individual $U$ vectors are linearly independent?
gct
  • 623
13
votes
5 answers

Characteristic polynomial of a matrix of $1$'s

I am trying to calculate the characteristic polynomial of the $n \times n$ matrix $A = \{ a_{ij} = 1 \}$ . Case $n=2$ : I obtained $p(\lambda)=\lambda^2-2\lambda$ . Case $n=3$ : I obtained $p(\lambda)=-\lambda^3+3\lambda^2$ . Case $n=4$ : I obtained…
Thom Kostova
  • 445
10
votes
1 answer

Cones of positive semidefinite matrices generated by matrices of rank $1$

Let $S_n$ be the space of real $n \times n$ symmetric matrices and let $S_n^+$ be the convex cone of positive semidefinite matrices in $S_n$. The extremal rays of this cone correspond to the positive semidefinite matrices of rank $1$. My problem is…
Max
  • 121
10
votes
1 answer

Convex hull of rank-$1$ matrices is the nuclear norm unit ball

Let $$A := \left\{ u v^T : u \in \mathbb{R}^m, v \in \mathbb{R}^n, \|u\|_2 = \|v\|_2 = 1 \right\}$$ I would like to show that $$\textrm{conv}(A) = B_* := \left\{ X \in \mathbb{R}^{m \times n}: \|X\|_* = \textrm{Tr} \left( \sqrt{XX^T} \right) \le 1…
guanton
  • 365
9
votes
1 answer

What properties does a rank-$1$ matrix have?

I have seen a lot of papers mentioning that a certain matrix is rank-$1$. What properties does a rank-$1$ matrix have? I know that if a matrix is rank-$1$ then there are no independent columns or rows in that matrix.
Matthew
  • 319
9
votes
2 answers

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 whether none exists (i.e., the positive linear…
M.A
  • 450
8
votes
3 answers

Calculating the minimal polynomial of a rank-$1$- matrix

Given the matrix $$ \begin{pmatrix} 1 & 1 & \dots & 1 \\ 2 & 2 & \cdots & 2 \\ \vdots & \vdots & \ddots & \vdots \\ n & n & \dots & n \end{pmatrix} $$ calculate the minimal polynomial. I know that the characteristic polynomial is $$ P_A(x) = x^{n-1}…
TheNotMe
  • 4,899
7
votes
4 answers

$\det(I+A)=1+\operatorname{Tr}(A)$ if $\operatorname{rank}(A)=1$

Let $A$ be a complex matrix of rank $1$ . Show that $$\det (I+A) = 1 + \operatorname{Tr}(A)$$ where $\det(X)$ denotes the determinant of $X$ and $\operatorname{Tr}(X)$ denotes the trace of $X$ . Any hint, please. I do not get how to combine the…
user598858
7
votes
4 answers

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 an obvious fact. And one more thing that came up…
User001
  • 1
6
votes
4 answers

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, selecting $x$ and $y$ as left and right…
Thomas Arildsen
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5
votes
1 answer

Frobenius and operator norms of rank 1 matrices

$\newcommand{\opnorm}[1]{\left\| #1 \right\|_{\mathrm{op}}} \newcommand{\norm}[1]{\left\| #1 \right\|}$ Suppose we have $X = x_1 x_2^\top \in \mathbb{R}^{n \times d}$ a rank-1 matrix which is non-symmetric in general. I want to either prove that $\|…
VHarisop
  • 4,130
5
votes
3 answers

Optimal symmetric rank-1 approximation

I want to find $\mathbf{x}$ that minimizes $\|A-\mathbf{x}\mathbf{x}'\|^2$ where $\|\cdot\|$ is Frobenius norm . Differentiating with respect to $\mathbf{x}$ and setting to $\mathbf{0}$, I get $$\mathbf{x}\mathbf{x}'\mathbf{x}=A \mathbf{x}$$ Any…
Yaroslav Bulatov
  • 5,639
4
votes
1 answer

Minimal "dominating" rank-$1$ matrix

Given a matrix ${\bf A} \in \Bbb R^{n \times n}$ , I would like to find a minimal rank- $1$ matrix ${\bf B} \in \Bbb R^{n \times n}$ such that the Frobenius norm of $ \| {\bf A} - {\bf B} \|_{\text{F}} $ is minimal under the constraint of $ a_{ij}…
Jiro
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