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Questions tagged [matrix-rank]

For questions regarding the rank of matrices in linear algebra.

In linear algebra, the rank of a matrix $A$ is the size of the largest collection of linearly independent columns of $A$ (the column rank) or the size of the largest collection of linearly independent rows of $A$ (the row rank). For every matrix, the column rank is equal to the row rank. It is a measure of the “nondegenerateness” of the system of linear equations and linear transformation encoded by $A$. There are multiple definitions of rank. The rank is one of the fundamental pieces of data associated with a matrix.

If $A$ is the matrix of a linear map $f$, then the rank of $A$ is equal to the dimension of the image of $f$.

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What is the rank of symmetric Pascal matrix?

Can two rows of a symmetric Pascal matrix (in general) be ever linear dependent? Is the rank of a symmetric Pascal matrix always equal to the number of rows (m)? I have a symmetric Pascal matrix where m=4 and n=4 and should count its rank.
Kozuch
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Line segment between two matrices and rank properties

Can any one help me by following question: Let assume we have two real-valued matrices $A,B\in R^{m\times n}$. Then, let define the matrix $G:=(A+t(B-A))(A+t(B-A))^T$ for a $t\in (0,1)$. Now, I am wondering, which conditions are sufficient to make…
Mehr
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Show that for any matrix $A_{m \times n}$ , the row rank and column rank are equal

Can somebody first please tell me what is the row rank and column rank of a matrix ? What is the relation of each with the rank of a matrix ? Any kind of explanatory proof would be very helpful , thanks ! Also , I am not looking for an intuitive…
Nikhil Kapoor
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Does ${\rm rk} AB \le {\rm min} ({\rm rk} A, {\rm rk} B)$ hold over a sum?

For matrices $A, B$, does the following rank (rk) inequality hold \begin{equation} {\rm rk} (AB + BA) \le {\rm rk}(AB) + {\rm rk}(BA) \le {\rm min}({\rm rk} A, {\rm rk} B) + {\rm min}({\rm rk} B, {\rm rk} A) = 2 ~{\rm min}({\rm rk} A, {\rm rk} B)…
Rob
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does the elementary transformation to a matrix change the rank?

Does the Rank decrease if the elements of a row are reduced to zero by elementary transformation ? I think Rank doesn't change because Rank of a matrix is the number of linearly independent row, which means the row which will be reduced to zero was…
Pranav Peethambaran
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problem of rank-nullity theorem with a given set

Let $A$ be an $m \times n$ matrix and $B$ be an $n \times p$ matrix. Let $$S=\left \{Bx\mid x\in \mathbb{R}^p \text{ and } ABx=0\right \}$$ Prove that $\dim(S)=\operatorname{rank}(B)-\operatorname{rank}(AB)$ What I thought is actually the…
mathshungry
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Matrix rank from rank of sub matrices

We know that if a matrix $\operatorname{rank}(A) \leq r$ where $A \in \mathbb{R}^{N \times M}$ then each submatrix $B_i \in \mathbb{R}^{n \times m}$ has $\operatorname{rank}(B_i)\leq r$ The opposite is true also if we take all $B_i \in \mathbb{R}^{n…
cdicle
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Mathematical representation of row independence for a matrix that is not full rank

I have the matrix $A$ below which is the permutations of $\{f_0,f_1,f_2,f_3\}$ \begin{equation*} A=\begin{bmatrix} f_0 & f_1 & f_2 & f_3 \\ f_0 & f_1 & f_3 & f_2 \\ f_0 & f_2 & f_1 & f_3 \\ f_0 & f_2 & f_3 & f_1 \\ f_0 & f_3 & f_1 &…
HVW
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Let $a,b,c$ are distinct integers. Then which among the following is the set of possible rank of $A$=

Let $a,b,c$ are distinct integers. Then which among the following is the set of possible rank of $$A = \begin{pmatrix} a^2 & b^2 & c^2 \\ a^5 & b^5 & c^5 \\ a^{11} & b^{11} &…
Largest Prime
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Solved. Rank of a singular matrix after reducing its range?

Suppose I have a linear system $Ax = b$. The matrix $A \in \mathbb{R}^{m \times m}$ has $Rank(A) = n < m$, so it is a singular matrix. Suppose I separate the variables into three groups: $x_i, x_-$ and…
Alberto Martín
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rank of $AB$ when $B$ has full column rank?

Suppose $B$ has a full column rank. Does it hold that $\operatorname{rank}(AB)=\operatorname{rank}(A)$? I found a similar post. But it does not prove or disprove the above statement.
mxdxzxyjzx
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How to prove rank(A+) is no more than rank(A)?

look here my friend. How to prove the following equation? Or give a counter-example. Thank you so much $$\text{rank}(A^+)\leq \text{rank}(A)$$ where $\text{rank}(A^+)$ represents the positive components of matrix $A$, e.g. $$\left[\begin{matrix}2 &…
guorui
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Matrix , rank of a matrix

Is it true that in equation $Ax=b$, $A$ is a square matrix of $n\times n$, is having rank $n$, then augmented matrix $[A|b]$ will always have rank $n$? $b$ is a column vector with non-zero values. $x$ is a column vector of $n$ variables. If not then…
y_159
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expected rank of a binary matrix with distribution

I want to know how to calculate expected rank of a binary matrix with distribution. I.e., the generated matrix are not randomly selected but with distribution. A more intuitive example is the LT code, it generates the matrix with the given degree…
desword
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Analysing Ranking Data

I'm trying to analyzing ranking data, a simple data set would be: I looked at some existing examples, one is survey monkey: .. but their equation is not clear, so just wondering whats the best way to do this when the options have a min and max…
Nimbuz
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