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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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Rank Equation for non-square Matrix

If I have a $n\times d$ matrix $A$, can we say $rank(A) + null(A) = rank(A^T) + null(A^T)$? Also, is $rank (A) + null(A) = max(n,d)$? Thanks in advance.
Lixiang Wei
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Matrix ranks and minors

Let $A$ be a $n \times n$. Let $P$ be a $A$ minor $(n-1) \times (n-1)$ with det $\neq 0$. Let $Q$ be a $P$ minor $(n-2) \times (n-2)$ with det $\neq 0$. Assume there are more than one $Q$ minor in $A$. If I find that all determinants $(n-1) \times…
xpeiro
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Matrix $C \in M_n(\mathbb{R})$ $\lambda_1=\lambda_2=0$ rank

If a have a matrix $C \in M_n(\mathbb{R})$ with $C^2=O_n$ and $n=2k+1$ that has at least to eigenvalues equal to $0$. Can I say from this that the $rank C\leq \frac{n-1}{2}?$
mathlearning
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Minimum number of non-zero entry in each row and column such that a matrix has maximum rank

Consider a square matrix of size N, such that each row and each column has exactly M non-zero positive elements. A simple example with N=3 and M=2 would be $ \left( \begin{array}{ccc} 0 & A1 & A2 \\ A3 & 0 & A4 \\ A5 & A6 & 0…
Michele Cotrufo
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Determining the rank of a matrix, am I missing something?

I have problems determining the rank of the following matrix. Note: Most probably it is me missing something, but I am sincerely confused by this: $$ \begin{matrix} 1 & 2 & -2 \\ 0 & 0 & 5 \\ 0 & 0 & -8 \\ \end{matrix} $$ Now, I…
l0ner9
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Why is the adjugate matrix the null matrix?

I struggle to understand why if $A\in M_n(\mathbb{C}) $, $\det A=0$ and $rank A \le n-2$, then $A^*=O_n$. Could you please tell me why this claim holds? My textbook offers no proof for this.
user69503
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Rank of matrix related

Is $ \rho (AA^*A)=\rho(A)$ always true? Here $\rho $ indicates rank of the matrix, and $A$ has entries from $\mathbb{C}$ I found that by multiplication inequality, $ \rho (AA^*A)\leq\rho(A)$. Now I am trying to disprove the strict inequality.…
user511110
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Rank of a matrix plus a number

Suppose to have a symmetric matrix $A$ with rank equal to $a$. Now let $$ A_1 = A + \frac{x}{2} $$ such that $x$ is a real number different from 0. Is it always true that rank of $A_1 = a+1$? Can you prove that? In which case this will be true?
Roberta
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generalization of [a matrix of rank at most 1 if and only if a matrix is a product of two vectors]

I know that a matrix is of rank $\le 1$ if and only if it is a product of two vectors. Then is there some generalization of this fact? Specifically, I am wondering that such sentence: a $m \times n$ matrix is of rank $\le 2$ if and only if it is a…
ddd
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Echelon form of a matrix

As far as I know each matrix has a unique echelon form. But sometimes I find more than one when I change my row operation. Why does it happen? Or is getting a unique echelon form just for row reduced echelon form?
dz420
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if anything can be said about rank of matrices if eigenvalues are known

If the eigenvalues of $A,M$ are all distinct $1,\lambda_2,\lambda_3$ and $1,m\lambda_2,m\lambda_3$, $B$ is any $3\times 3$ matrix with positive entries(0 is allowed), If the rank of $K=[B\hspace{0.5cm} AB\hspace{0.5cm} A^2B]$ is $3$ can we say…
Myshkin
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Finding the rank

Let A be a $3\times4$ matrix of rank $2$ for some real numbers $a$ and $b$. What is the value of $b$? where, $$A = \begin{pmatrix} 1 & 1 & 2 & 2 \\ 1 & 1 & 1 & 3 \\ a & b & b & 1 \\ \end{pmatrix}$$ How to approach…
math is fun
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If it's true for $\textrm{rank}(A)=\textrm{rank}(A^2)$

I am trying to prove this and found this may be not true in general.Does anyone have any ideas about the counterexample for this statement?
wenyuan wang
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Elementary tranformations in matrix

Very often we use elementary transformations to find rank of matrices. Which of the following transformations are correct? I guess that only 1 is correct and 2, 3 are wrong. Am I right? $R_2\to R_2-kR_3$ $R_2\to 2R_2-kR_3$ $R_2\to -R_2-kR_3$
user3575652
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Find all values of $x$ for which $\text{rank}(A) = 2$.

$\{\{-1,1,0,7\},\{-2,2,x,-7´\},\{-1,3,2,0\}\}$ I have tried this and I am getting a rank of $3$ for all values. Ia there any value that actually gives a rank of $2$.
Padraig O Connor
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