# Questions tagged [rank]

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### Spot redundant equations within nonlinear system of equations

Is there a general procedure to detect if in a system of m-nonlinear equations (also non polynomial) of n-unknowns some of the equations are redundant? Can the rank of the Jacobian matrix tell me ...
• 23
112 views

### Rank of a double-precision augmented matrix

Let $A$ be a matrix with real entries, and let $A_+$ be $A$ augmented by a single column. From linear algebra we know \operatorname{rank}(A_+) = \operatorname{rank}(A) \hspace{10pt} ...
• 255
230 views

I have a matrix $A\in\{0,1\}^{d\times n}$ and $rank(A)=d,d<n$, and another matrix $X\in \mathbb{R}^{d\times n}$, but I do not know the rank of $X$. What can we say about the rank of their Hadamard ...
• 89
1 vote
379 views

### Rank filter on an nXm array using python

I would like to apply a rank filter on an nXm numpy array. Let's say I have this array: ...
• 121
652 views

### Rapidly determining whether or not a dense matrix is of low rank

In a software project that I'm working on, certain computations are vastly easier for dense low-rank matrices. Some problem instances involve dense low-rank matrices, but they're given to me in full, ...
• 18.7k
1k views

### Quadratic programs with rank deficient positive semidefinite matrices

Let $A$ be a $n\times n$ square symmetric matrix. In addition, $A\succeq0$ and $\mathrm{rank}(A)<n$. This means that all eigenvalues are non-negative, but also that there are some zero eigenvalues. ...
1 vote
361 views

• 383
1 vote
251 views

### Fast counting of all submatrices of a binary matrix with a full column rank

I have a binary full-rank matrix of size, say, $25 \times 50$. I need to count how many subsets of its columns form matrices with a full column rank, i.e. all the columns in the subset are linearly ...
173 views

### Minimize the number of unique elements in a vector

I was wondering if there is a simple or known way to minimize the number of unique elements in a decision variable (vector). Note that I'm not asking for minimization of nonzero elements (rank ...
• 21
2k views

### How to compute the rank of a large sparse matrix in MATLAB

I am interested in computing the ranks of fairly large, the largest being of magnitude $10^6$ x $10^6$, sparse matrices whose entires are all 0, 1, or -1. I have been trying to use Matlab to ...
• 171
1 vote
262 views

### Diagonalize a circulant-plus-rank-one matrix

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 $A$...
149 views

### Rank of image intensity matrix

I've been reading a paper about using Matrix Completion for Photometric Stereo but I am having some troubles in section 2.2 trying to understand why irrespective of the number of pixels and the number ...
• 1,029
1 vote
103 views

### Sparse matrix factorization of a rank deficient matrix by decomposition into linearly independent components

I've got a little conjecture I need to prove for a theoretical result related to causal Bayes net search with latent variables under sparsity constraints. If you're interested in the application ...
• 11
2k views

### Computational complexity and implementation of UDU Modified Cholesky Rank 1 Update

I am attempting to increase the performance of a legacy Kalman Filter implementation. The state covariance is factored in terms of UDU, i.e. $\mathbf{P} = \mathbf{U}\mathbf{D}\mathbf{U}^T$. Many ...
• 802
### Writing Real Symmetric Matrices as Linear Combination of Rank One Symmetric Terms $uu^T$
Given a real symmetric matrix $M$, ostensibly of "low rank", efficiently find an expression $M = \sum \alpha_i u_i u_i^T$ using the number of terms rank($M$). A 2011 StackOverflow Question Dense ...