Questions tagged [matrix-rank]

For questions regarding the rank of matrices in linear algebra.

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A Theorem that proves $rank(T) = \dim \operatorname{Im}(T)$

I'm looking for someone who knows a theorem who proves that this two definitions of rank of Linear Transformations/Matrices are equivalent. $rk(T) = \dim \operatorname{Span}( \{(a_{i1}, \dots, a_{in}),...
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$\mathrm{rank}(I_n-BA)=\mathrm{rank}(I_n-AB)$ [duplicate]

Is it always true for $A,B \in \mathcal{M_{n}}(\mathbb{C})$ that $$\mathrm{rank}(I_n-BA)=\mathrm{rank}(I_n-AB)\ ?$$ I'm looking for a matrix theoretic proof, not involving dimension theory. Thanks in ...
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Does it always hold that $n-\operatorname{rank}(A)=\operatorname{rank}(I_n-BA)-\operatorname{rank}(A(I_n-BA))$

Let $A,B \in \mathcal{M_{n}}(\mathbb{C})$. Does it always hold that $$n-\operatorname{rank}(A)=\operatorname{rank}(I_n-BA)-\operatorname{rank}(A(I_n-BA))\ ?$$ Attempt: Begin by noticing that $$\ker(A)...
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Singular points of the commutator map

I would like to determine the singular points and the singular values of the polynomial map $f:\mathbb{R}^8\to\mathbb{R}^5$ \begin{multline*} f(x_1,\ldots,x_8)=(x_1x_4-x_2x_3,x_5x_8-x_6x_7,x_2x_7-...
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Rank of the concatenation of two matrices [closed]

Let $A, B \in \Bbb R^{n_1 \times n_2}$ and let $C := [A\,\,\,B]$ be a $n_1 \times 2n_2$ matrix whose first $n_2$ columns are $A$ and whose remaining columns are $B$. Is it true in general that $\mbox{...
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Any thought on that rank similar object?

Context : This problem arises on some exploration around Wyner's common information in information theory and the related minimization problem. Problem : Let $A$ be a $m\times n$ real matrix. Its ...
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Comparing the ranks over $\mathbb{Q}$ and $\mathbb{C}$ of a matrix of $\mathcal{M}_n(\mathbb{Z})$

Let $A$ be a matrix with coefficients in $\mathbb{Z}$. What can be said about the rank of $A$ over $\mathbb{Q}$ and $\mathbb{C}$? Is the rank of $A$ over $\mathbb{C}$ less than the rank of $A$ over $\...
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$\operatorname{rank}(A+B)\geq|\operatorname{rank}(A)-\operatorname{rank}(B)|$ [duplicate]

Question: enter image description here Solution: enter image description here $\operatorname{rank}(A+B)\geq|\operatorname{rank}(A)-\operatorname{rank}(B)|$ I have recently started learning linear ...
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Transform stacked matrix into block-diagonal form

Consider two matrices $A$ and $B$ that get stacked to form a (tall) matrix $J$, $$ J = \left[\begin{array}{l} A\\ B \end{array} \right]. $$ Assume that $\text{rank}(J) = \text{rank}(A) + \text{rank}(B)...
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Determinant of a $2 \times 2$ block matrix whose diagonal blocks are skew-symmetric

Let $$A = \begin{pmatrix} B & C \\ C' & D\end{pmatrix}$$ be an odd order matrix. If blocks $B, D$ are skew-symmetric matrices, then $\det A=0$. My attempt Without losing generality, we assume ...
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Is the product of two independent random matrix full rank?

The elements in matrix $A$ and $B$ are independent and absolute continous random variable. Is $AB$ be full rank with probability one? First, $A$ and $B$ must be full rank with probability one. I try ...
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Possible ranks of matrix if $\operatorname{rank}(AB)$ = $\operatorname{rank}(BA)$

If matrix $A \in M_n(R)$ is defined such as for each $B \in M_n(R)$ we have $\operatorname{rank}(AB) = \operatorname{rank}(BA)$ find all possible ranks of $A$. So obviously, $I$ and $0$ are both one ...
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True or False: There is a $6\times 6$ matrix $A$ with $\text{Rank}(A)=4$ and $A^3 =0$

I understand how to do it if the question changed $A^3$ to $A^2$, because then you can just use the rank–nullity theorem. $\text{Rank}+\text{Nullity}=6$, $\text{Rank}=4$ so $\text{Nullity}=2$ so of ...
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Set of measures on the simplex of matrices that take support on rank 1 matrices has a minimal element

Let $M=\{ A\in\mathbb R^{n\times m} : \forall i,j~A_{i,j}\geq 0,\sum_{i,j} A_{i,j} = 1 \}$ be the simplex of $n\times m$ matrices, it was shown in a previous question that the set $R$ of rank $1$ ...
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$B^2=B$ implies $rank(AB-BA) \leq rank(AB+BA)$ [duplicate]

$A,B$ is square matrix. I have to prove that $B^2=B$ implies $rank(AB-BA)\leq rank(AB+BA)$ I know $(I-B)(AB+BA)=AB-BAB=(AB-BA)B$. But I'm not sure what to do next.
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Determine rank of $ A = \begin{bmatrix} 2 & 1 & -2 & 1 \\ 4 & 1 & -2 & -3 \\ 1 & -1 & 2 & -3 \\ 2 & 2 & -4 & -5 \\ 3 & 1 & -2 & 2 \end{bmatrix}$

Could you give me your feedback ? I've verified with https://matrix.reshish.com/rankCalculation.php but maybe there are things that could be done differently Determine the rank of the following ...
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When is $Rank(A+B)=Rank(A)$

$A$ and $B$ are two square symmetric matrices each of order $n×n$ and $Rank(A)=Rank(B)=n-1$. Also $A\textit 1=\textit 0$ and $B\textit 1=\textit 0$. Let $C=A+B$. I want to know about $Rank(C)$, ...
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Rank of matrices and their block form

My 2 Questions: Any matrix can be brought into block form however, why does $A$ and $B$ need to have the same rank? It follows further that Any $m \times n$ matrix is equivalent to the block matrix ...
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What the rank of a matrix with the elements of one column to be infinity? [closed]

Suppose the $m\times m$ real matrix $A$ is positive definite, it is without doubt that $\mathrm{rank} (A) = m$. Now, if all the elements in the first column and first row are assigned to be infinity, ...
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Is the set of rank 1 matrices positive matrices that sum to $1$ closed?

This question arises from my exploratory research on causality. Let $M=\{ A\in\mathbb R^{n\times n} : \forall i,j~A_{i,j}\geq 0,\sum_{i,j} A_{i,j} = 1 \}$ be the simplex of matrices of size $n\times n$...
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The similarity of a $n\times m$ matrix

Let $F$ be a field and let $A$ be a $n\times m$ matrix which has rank $r$. By the dot number 9 in the section Proposition, there exists an invertible $m\times m$ matrix $X$ and an invertible $n\times ...
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If entries at pivot position are zero then the column is a linear combination of the previous columns?

In the 7th lecture of Gilbert Strang's Linear Algebra course, he does row operations on a matrix (m x n) to convert it to echelon form. He found the first k pivot elements; for k+1 th pivot element, ...
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Rank of permutations matrix

Q. Let $u=\left(u_{1}, u_{2}, \ldots, u_{n}\right)$ be a fixed non-zero vector in $\mathbb{R}^{n}$ such that $u_{1}+u_{2}+\cdots+u_{n}=0$, and let $A$ be the $n ! \times n$ matrix whose rows are the $...
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Do matrices over noncommutative division rings have well-defined ranks?

It is known that the row and column rank of any matrix over a field are the same and their common value is simply called the rank of the matrix. Now, for any $m$-by-$n$ matrix $A$ with entries in a ...
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Rank of the matrix $P=X(X^TX)^{-1}X'$

$X$ is an $n×p$ matrix, with $Rank(X)=r$. Then what is the Rank of the matrix $P=X(X^TX)^{-1}X'$? Is it always true that $Rank(X)= Rank(P)$? Is it true when $X$ is of full column rank, i.e. $p=r$? I ...
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intuitive explanation why rank equal dimension of column space

A nxn, x and B are both nx1 Ax=B First show that if rows are all independent then column are all independent: Intuitively: n independent rows (representing n “non-parallel” equations) in n unknowns ...
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3 votes
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Why $\operatorname{rank} AB≠\operatorname{rank} BA$?

Let A,B be two square matrices, say of size n, then why $\operatorname{rank} AB≠\operatorname{rank} BA$? Just need valid reasoning.I just know that above questions has answer 'No'.
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Kernel of a $2 \times 2$ block matrix with diagonal blocks zero

We have a $2 \times 2$ block matrix M, in which the diagonal blocks are $0$. The sizes of the upper rigt block is $N_B \times N_A$ and the size of the lower left block is $N_A \times N_B$, where we ...
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Does $A=B$ imply that the rank of $A$ equals the rank of $B$?

I am trying to understand a poof of the clubs of oddtown theorem, and I am stuck at this step: If $A$ is an $m \times n$ matrix and $AA^T=I_m$, then the rank of $AA^T$ is AT LEAST $m$. Now if $A=B$ ...
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Prove that if $A$ and $B$ are $n\times n$ matrices, $A$ is invertible and $BAB=0$ then ${\rm rank}(B) \leq n/2$

Let $A, B$ are $n\times n$ matrices and $A$ is invertible. I need to show that if $BAB=0$ then ${\rm rank}(B) \leq \frac{n}{2}$. How can I do it? I can prove that $$0 = {\rm rank}(BAB) \leq {\rm rank}...
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Proof: Rank of block of matrix is smaller than rank of matrix

Let $$A = \begin{bmatrix} A1 \\ A2\end{bmatrix}$$ be a matrix with real entries Then proof $Rank(Ai) ≤ Rank(A)$ for $i = 1, 2$ I am attaching my solution sheet: Solution Can someone help me ...
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The rank of an antisymmetric matrix

Let $A$ be an antisymmetric matrix of order $(2n+1)\times(2n+1)$, whose non-zero elements are either $1$ or $-1$, and that has no zero rows. I know that rank$(A)<2n+1$ since it is not invertible. ...
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Prove that the augmenting skew symmetric matrix has full rank.

Suppose that $\vec{v}$ is a unit vector where $v_3 \neq 0$, and $M(\vec{v})$ be skew symmetric matrix written as \begin{equation} M(\vec{v})=\begin{bmatrix} 0 & -v_3 & v_2 \\ v_3 & 0 & ...
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How can i demonstrate that rank(AB-BA)=rank(AB+BA)?

There is a natural $n,$ such that $n≥2$ where $A, B$ are square matrices of order n, idempotents, and that det($A$$-$$B$)$≠$$0$ I tried to use the idempotence propriety, so I thought about starting ...
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Is my proof about the rank of a matrix correct?

The problem: Let $A\in\mathbb{K}^{m\times n}$ be a matrix. Show that $\text{rank}({A})$ is identical to the largest number $k\in\{1,2\dots,\min(m,n)\}$ such that the subdeterminant with $i_1<i_2<...
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2 votes
2 answers
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A question about the rank of some matrix

Let $P$ be a $s \times s$ matrix and $Q$ be a $s \times r$. Assume that $\mbox{rank} (P|Q)=s$, can we find a $r \times s$ matrix $R$ such that $\mbox{rank} (P+QR)=s$?
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3 votes
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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 ...
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Prove that $\text{rank}(\mathbf{I}-\mathbf{X}(\mathbf{X}^\intercal\mathbf{X})^{-1}\mathbf{X}^\intercal)=n-k-1$, where $\mathbf{X}$ is $n\times k+1$

This is required to show that $\text{SSR}/\sigma^2$ is $\chi^2(n-k-1)$, where SSR is the residual sum of squares $(\mathbf{y}-\mathbf{X}\boldsymbol{\hat{\beta}})^\intercal(\mathbf{y}-\mathbf{X}\...
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Existent of matrix sequence such that $AX_nB$ is full rank

Suppose I have two matrices $A\in \mathbb{R}^{p\times m}$, $B\in \mathbb{R}^{n\times r}$, where $m\geq n\geq p\geq r$ and both $A,B$ have full rank. Given a low-rank matrix $X\in \mathbb{R}^{m\times n}...
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Is it possible to find the minimal polynomial of a matrix knowing its rank?

We know that any $n \times n$ matrix $X$ with $rank(X)=1$ can be written $u(^Tv)$ where $u, v$ are vectors and $^TA$ is the transpose of $A$. This leads to $X^2=u(^Tv)u(^Tv)=u(^Tvu)(^Tv)=cu(^Tv)$, ...
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rank of a matrix and its transpose

Let $A$ is a $n \times n$ matrix, and $B$ is a $n \times m $ matrix and $m\leq n$. If $rank[A\quad B]=n$, can we get $rank[A\quad B*B']=n$? I try to prove that $B$ and $B*B'$ have the same range space....
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Conditions for which a collection of i.i.d. random vectors has full rank almost surely

Let $X_1, \ldots, X_N:(\Omega, \mathcal F, \mathbb P) \to \Delta \subseteq \mathbb R^n$ be i.i.d. random vectors with $N>n$. Here $\Delta$ is the probability simplex. Let $$ A := \{\omega \in \...
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For any $A\in M_n(\mathbb{R})$, there is a $B$ s.t. $\operatorname{rank}(A)+\operatorname{rank}(B)=n$ and $AB=0$.

This is a qualifying exam question which I have little experience approaching. This is certainly an exercise in linear algebra so I would like to obtain a method in solving similar questions. Let $...
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Two questions about Cayley-Menger determinant

Let $\mathbf p_1, \ldots,\mathbf p_n$ be $n$ points in $\Re^d$ and let $k$ be the maximum number of affinely independent points among them. Consider the following matrix $D$ whose determinant is known ...
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Rank of a complex matrix

I was reading a text book, and encountered the following matrix: $H = h[e_r(\Omega_{r1})$ $e_r(\Omega_{r2}]$. Where, $e_{r}(\Omega_{rk})$ is a colomn vector with entries as $e^{-j2\pi k\Omega_{rk}}$. ...
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Is the rank $A+\sqrt{2} B$ greater than the rank of $A$?

Let $A$ and $B$ be two matrices of same dimensions with rational coefficients. Is $\text{rank } A+\sqrt{2} B \ge \text{rank } A$ and $\text{rank } A+\sqrt{2} B \ge \text{rank } B$? More generally, is ...
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Some questions of the proof related to Low-rank approximation on wikipedia

i lost the way when i read the proof of Eckart–Young–Mirsky theorem (for Frobenius norm) recorded on wikipedia(enter link description here). i firstly want to know in the following logitic of the ...
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The matrix with entries $\pm 1$ is surjective?

Consider matrix $A$ of size $n\times 2^n$ such that each column is a vector in $\{-1, +1\}^n$. It is clear that there are exactly $2^n$ distinct columns. I have the impression that the linear map ...
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2 votes
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I have a (wrong?) proof that a matrix with a unit eigenvalue can't have a zero eigenvalue. Where is the illegal step?

We know that a square matrix (for example in $C$), can have eigenvalues equal to 0 or 1 simultaneously. For example: $$ A= \begin{pmatrix} a & b\\ c & d \end{pmatrix} = \begin{pmatrix} 1 &...
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Relationship between PSD matrix rank and its convexity

I know that the matrix $W=vv^T$ is PSD and of rank 1. I also know that the set of rank-1 matrices is not convex. My question may be trivial but I would appreciate your feedback. If $W$ is PSD, it ...
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