All Questions
Tagged with matrix-decomposition matrix-rank
128
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Proving that the rank of the following matrix is $6$.
In my research work I have come across a matrix which has the rank equals to $6$. I begin defining my problem as follows: Let $P \in \{0,1\}^{7 \times 7}$ denote the right shift matrix defined by
$ P =...
- 49
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27
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A low-rank approximation problem with rank constraints
I am seeking a solution or some ideas to address the following problem:
$$
\begin{aligned}
&\text { minimize }_{\widehat{A}, \widehat{B}} \quad\|A-\widehat{A}\|_2 + \|B-\widehat{B}\|_2 \\ &\...
- 11
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Detect linearly dependent columns from a full-row rank matrix.
Let $A\in \mathbb{R}^{m\times n}$, with $m\leq n$, be a full-row-rank matrix. Then, there exist a collection of $n-m$ columns in $A$ that are linearly dependent on the other columns. What is the most ...
- 372
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Degrees of freedom of an $r$-ranked tensor?
I'm trying to determine the degrees of freedom for parameters of a tensor in shape $(J_1,\dots,J_D)$ and with rank $r$, where "rank" refers to the smallest number of rank-1 tensors whose sum ...
- 213
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Given a singular matrix $B$ and a result $C=A\times B$, find matrix $A$ over finite fields.
The problem is a conituation of this problem, but over finite fields.
In the context of a finite integer field, particularly when all entries in matrices $A, B$, and $C$ are drawn from a finite ...
- 95
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28
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Interpretation of QR "values" (a la singular values)?
Let $A\in\mathbb{R}^{m\times n}$ with $m>n$ and $\text{rank}A=n$.
There exists $\hat{Q}\in\mathbb{R}^{m\times n}$ with orthonormal columns and $\hat{R}\in\mathbb{R}^{n\times n}$ upper triangular ...
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105
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Specific basis for the space of symmetric matrices
Consider the space of symmetric matrices $symm(M)$ over reals of dimension $n \times n$. It is clear that there is a straightforward basis for this space where for any $i \ge j$ $M_{ij}(m,n) = 1$ if $...
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Let A be an m by n matrix and let B be an n by p matrix. Show that rank A + nullity A = n is the special case of rank AB + dim(null A ∩ col B)=rank B.
Let $A$ be an $m \times n$ matrix and let $B$ be an $n \times p$ matrix. Show that $\text{rank } A + \text{nullity } A = n$ is the special case of $\text{rank } AB + \dim(\text{null } A \cap \text{col ...
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134
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Rank 1 matrix with given anti-diagonal sums
I'm looking for a matrix that is of rank 1 but has specified anti-diagonal sums. To give a few examples:
It is easy to make a rank 3 matrix which has anti-diagonals that sum to 1:
\begin{bmatrix}
1 &...
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83
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Matrix product that preserves the rank
Consider the matrix product $A=B\cdot C$, where $A$ is a $n\times n$ matrix, $B$ is a $n\times m$ matrix ($n<m$), and $C$ is a $m\times n$ matrix.
Suppose that $\rho(B)=n$ (i.e., $B$ has full row ...
- 513
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64
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Product of matrices with only one nonzero entry on each column
Let $R$ be a full-rank $n \times m$ rectangular matrix with entries in a DVR, with $n \leq m$ (less rows than columns) and we may assume $n < m$ if useful. Is there an $m \times m$ invertible ...
- 566
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Decomposition of a matrix in $\mathbb{M}_n(\mathbb{C})$ into the sum of two rank-1 diagonalizable matrices.
I would like to solve the following problem.
Let $n \in \mathbb{N}$ such that $n \geq 2$.
Let $\mathrm{M} \in \mathfrak{M}_{n}(\mathbb{C})$. Under what condition(s) do there exist two diagonalizable ...
- 106
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Invertibility of the symmteric matrix observed under least squaress solution
I am having trouble in proving the following statement
Let $X$ be a $K \times d$ matrix with rank $d$ and let $W$ be a diagonal matrix with $d$ non-zero positive elements on the diagonal, that is rank ...
- 477
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100
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Low rank approximation with MSE approach
Suppose that we have a matrix $\mathbf{A}\in\mathbb{R}^{n\times p}$. If we want to find an approximation of $\mathbf{A}$ with maximum rank $k$, which is denoted as $\tilde{\mathbf{A}}$, is it possible ...
- 11
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Is there a name for this kind of matrix decomposition?
A square matrix $\mathbf{A}$ (of size $a\times a$) is not full rank and I want to decompose it as $\mathbf{A}=\mathbf{B}\mathbf{B}^\top$ where $\mathbf{B}$ is an $a\times(a-1)$ matrix of the $a-1$ ...
