Questions tagged [matrix-rank]
For questions regarding the rank of matrices in linear algebra.
2,208
questions
0
votes
1
answer
46
views
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}),...
-1
votes
1
answer
47
views
$\mathrm{rank}(I_n-BA)=\mathrm{rank}(I_n-AB)$ [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 ...
- 189
0
votes
2
answers
50
views
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 $$\ker(A)...
- 189
1
vote
0
answers
52
views
+50
Singular points of the commutator map
I would like to determine the singular points and the singular values of the polynomial map $f:\mathbb{R}^8\to\mathbb{R}^5$
\begin{multline*}
f(x_1,\ldots,x_8)=(x_1x_4-x_2x_3,x_5x_8-x_6x_7,x_2x_7-...
- 244
0
votes
1
answer
29
views
Rank of the concatenation of two matrices [closed]
Let $A, B \in \Bbb R^{n_1 \times n_2}$ and let $C := [A\,\,\,B]$ be a $n_1 \times 2n_2$ matrix whose first $n_2$ columns are $A$ and whose remaining columns are $B$. Is it true in general that $\mbox{...
- 115
0
votes
0
answers
21
views
Any thought on that rank similar object?
Context : This problem arises on some exploration around Wyner's common information in information theory and the related minimization problem.
Problem : Let $A$ be a $m\times n$ real matrix. Its ...
- 3,975
0
votes
0
answers
12
views
Comparing the ranks over $\mathbb{Q}$ and $\mathbb{C}$ of a matrix of $\mathcal{M}_n(\mathbb{Z})$
Let $A$ be a matrix with coefficients in $\mathbb{Z}$. What can be said about the rank of $A$ over $\mathbb{Q}$ and $\mathbb{C}$? Is the rank of $A$ over $\mathbb{C}$ less than the rank of $A$ over $\...
- 1,203
1
vote
1
answer
54
views
$\operatorname{rank}(A+B)\geq|\operatorname{rank}(A)-\operatorname{rank}(B)|$ [duplicate]
Question:
enter image description here
Solution: enter image description here
$\operatorname{rank}(A+B)\geq|\operatorname{rank}(A)-\operatorname{rank}(B)|$
I have recently started learning linear ...
- 23
1
vote
1
answer
26
views
Transform stacked matrix into block-diagonal form
Consider two matrices $A$ and $B$ that get stacked to form a (tall) matrix $J$,
$$
J = \left[\begin{array}{l}
A\\
B
\end{array}
\right].
$$
Assume that $\text{rank}(J) = \text{rank}(A) + \text{rank}(B)...
- 135
1
vote
2
answers
37
views
Determinant of a $2 \times 2$ block matrix whose diagonal blocks are skew-symmetric
Let $$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 ...
- 23
1
vote
0
answers
11
views
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 ...
- 11
0
votes
0
answers
45
views
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 ...
- 103
2
votes
1
answer
73
views
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 ...
- 21
0
votes
0
answers
11
views
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$ ...
- 3,975
0
votes
0
answers
44
views
$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.
- 9
1
vote
1
answer
73
views
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 ...
- 103
1
vote
0
answers
74
views
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)$, ...
- 853
1
vote
0
answers
52
views
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 ...
- 63
0
votes
1
answer
29
views
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, ...
- 13
0
votes
1
answer
40
views
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$...
- 3,975
0
votes
1
answer
20
views
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 ...
- 401
0
votes
1
answer
25
views
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, ...
- 3
0
votes
1
answer
68
views
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 $...
1
vote
0
answers
24
views
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 ...
- 7,965
0
votes
1
answer
51
views
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 ...
- 853
0
votes
0
answers
22
views
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 ...
3
votes
1
answer
81
views
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'.
1
vote
0
answers
42
views
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 ...
- 11
0
votes
1
answer
47
views
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$ ...
- 2,147
0
votes
1
answer
51
views
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}...
- 46
0
votes
1
answer
28
views
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 ...
1
vote
1
answer
32
views
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.
...
- 33
0
votes
0
answers
16
views
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 & ...
- 13
0
votes
1
answer
74
views
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 ...
- 29
0
votes
0
answers
30
views
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<...
- 55
2
votes
2
answers
89
views
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$?
- 257
3
votes
1
answer
47
views
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 ...
- 31
1
vote
1
answer
20
views
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}\...
- 157
0
votes
1
answer
39
views
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}...
- 27
0
votes
0
answers
36
views
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)$, ...
- 139
1
vote
1
answer
44
views
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....
- 951
0
votes
0
answers
18
views
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 \...
- 1,153
5
votes
4
answers
164
views
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 $...
0
votes
0
answers
30
views
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 ...
0
votes
0
answers
31
views
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 $e^{-j2\pi k\Omega_{rk}}$. ...
- 119
1
vote
1
answer
40
views
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 ...
- 21
0
votes
0
answers
28
views
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 ...
- 83
1
vote
1
answer
56
views
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 ...
- 1,208
2
votes
1
answer
70
views
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 &...
0
votes
0
answers
14
views
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 ...
