# Questions tagged [matrix-rank]

For questions regarding the rank of matrices in linear algebra.

1,311 questions
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### Absolute convex hull of rank 1-correlation matrices?

Does there exist a ''universal'' constant, $c > 0$ say, such that for any(!) $k \in \mathbb{N}$ every(!) $k \times k$-correlation matrix $\Sigma$ can be written as $\Sigma = c\Theta$, where $\Theta$...
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### Can $A + uv^T$ be non-singular if A(order n) has rank n-1

Suppose $A$ is a matrix of order $n$ with rank n-1, $u$ and $v$ are $n$-vectors. Can $A + uv^T$ be nonsingular, if yes then find such $u$ and $v$. First I wrote Rank$(A + uv^T) <=$ Rank$(A)$ + ...
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### What is the angle between a $A_{3x3}$ with $Rank(A)=2$ and $A^T$

I need to solve this problem: For Matrix $A_{3x3}$ with $Rank(A)=2$. If Matrix A is Transposed and its elements are the same as elements of Matrix B. What is the angle of rotation from A to B?
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### How far from a matrix the rank does not decrease?

Let $A$ be a real $n \times n$ matrix, and suppose that $\text{rank}(A)=k$. Consider $$r_0=\sup \{r>0 \,\, | \,\, |B-A| \le r \Rightarrow \text{rank}(B) \ge k \}.$$ Does $r_0$ depend only on ...
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### If $G$ is NOT full rank matrix, does minimum singular value $\sigma$ Zero? [closed]

I quite sure about in but didn't found a prove. Is it true?
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### What is a rank-1 tensor? What is the meaning of rank in this context?

I feel like different sources use the term "rank" differently, which is perhaps leading to my confusion. When I think of rank I think of number of linearly independent columns/rows, number of pivots ...