Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [matrix-rank]

For questions regarding the rank of matrices in linear algebra.

0
votes
0answers
7 views

Absolute convex hull of rank 1-correlation matrices?

Does there exist a ''universal'' constant, $c > 0$ say, such that for any(!) $k \in \mathbb{N}$ every(!) $k \times k$-correlation matrix $\Sigma$ can be written as $\Sigma = c\Theta$, where $\Theta$...
0
votes
1answer
22 views

$Ker(A)=\{0\} \Leftrightarrow rank(A)=n$

Let $A: n\times n$, I want to prove that if $N(A)=\{0\}$ if, and only if, $A$ has $n$ linearly independent columns. Important: You cannot use $dim(N(A)) + dim(R(A)) = n$. I thought of doing so, as $...
1
vote
0answers
41 views

Rewrite a condition on a $3\times3$ matrix

Consider the $3\times 3$ matrix $$ A\equiv \begin{pmatrix} \mu_1-\mu_1'& \mu_1-\mu_1'-c & \mu_1-\mu_1'-c-d\\ \mu_1+a-\mu_1'& \mu_1+a-\mu_1'-c & \mu_1+a-\mu_1'-c-d\\ \mu_1+a+b-\mu_1'&...
0
votes
0answers
26 views

Some doubts regarding the evaluation of the rank of a matrix

Here is a list of very novice questions that came across while studying: Suppose $A$ is an $m \times n$ matrix. Is the rank of $A\leq \min\{m,n\}$? Attempt: The "rank" of a matrix gives us the idea ...
0
votes
0answers
25 views

Rank of a Complex matrix in terms of its real and imaginary parts

Let $A\in\mathbb{C}^{n\times n}$ be a complex matrix (i.e., with complex entries). We separate the real and imaginary parts of $A$ as follows: $A=A_1+iA_2$ where $A_1,A_2\in\mathbb{R}^{n\times n}$. Is ...
0
votes
0answers
27 views

Special Block Matrix

Let $X,Y\in\mathbb{R}^{n\times n}$ be real square matrices. Let construct the $2n$-by-$2n$ block matrix $$ Z=\begin{bmatrix} X&Y\\ -Y&X \end{bmatrix}. $$ Do the matrices with the block ...
1
vote
1answer
42 views

How to prove the following rank equality?

Let $A,B\in M_{n}$. Show that $$\mbox{rank}\,(A-ABA)=\mbox{rank}\,(A)+\mbox{rank}\,(I_{n}-BA)-n$$ I have showed one part by using Sylvester rank inequality. How to prove the equality case?
0
votes
1answer
22 views

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 $rank C\leq \frac{n-1}{2}?$
3
votes
1answer
29 views

$a_k = rank(A^{k+1}) - rank(A^k)$ is increasing

$a_k = rank(A^{k+1}) - rank(A^k)$ is increasing. This is equivalent to $$ rank(A^{k+1}) + rank(A^{k-1}) \geq 2 rank(A^k) $$ which is a consequence of the Sylvester inequality: $$rank(XZY) + rank(Z) ...
0
votes
0answers
28 views

Rank of covariance matrix

I am having a problem with rank deficiency in a covariance matrix. I have a data-set of M variables and N observations, M>N. Calculating the singular value decomposition of the data-sets covariance ...
0
votes
0answers
23 views

Convex hull of {-1, 1} rank-1 matrices?

Consider set $\mathbb{R}^{m\times n}$ of $m \times n$ matrices. I'm particularly interested in properties of polytope $P$ defined as a convex hull of all {-1,1} matrices of rank 1, that is, $$ P = \...
0
votes
0answers
14 views

Pseudo determinant of a non negative matrix (singular matrix here ).

This wikipedia article gives the following formula for finding the pseudo determinant of a matrix. $$ |A|_+ = \lim\limits_{\alpha\to0} \frac{|A+\alpha I|}{\alpha^{n-\mathrm{rank}(A)}} $$ where $A$ is ...
1
vote
4answers
45 views

If rank of a given matrix of order $3 \times 4$ is $2$ then the value of $b$ is

Q) Suppose the rank of the matrix $\begin{pmatrix} 1 &1 &2 &2 \\ 1&1 &1 &3 \\ a&b &b &1 \end{pmatrix}$ is $2$ for some real numbers $a$ and $b$. Then $b$ equals $...
1
vote
1answer
25 views

Equality of Ranks

Let $A\in \mathbb{F}^{m×n}$. The row rank of $A$ is the dimension of the row space of $A$. Show that the row rank of $A=$ row rank of its RREF. Show that the row rank of $A$ is equal to rank of $A$. ...
3
votes
1answer
24 views

The “range” and “image” of a transformation refer to the same thing, right?

I'm hoping this is true because I've been told that "the rank of a transformation is the dimension of its image", and also that "the rank of a transformation is the dimension of its range". This doesn'...
1
vote
1answer
16 views

Rank of a matrix over field extension.

Let $A$ be a matrix over a field $\mathbb{F}$ and $\mathbb{K}$ be a field extension of $\mathbb{F}$. As I know that characteristic and minimal polynomial of $A$ over $\mathbb{F}$ and $\mathbb{K}$ are ...
0
votes
1answer
43 views

Two matrices whose product is equal to the identity matrix

I need to multiply two matrices one we call P type 1x4 another called Q type 4x1 I cannot find values that will give me an identity matrix as a result when I multiply PQ together. Where zeros are ...
0
votes
1answer
26 views

Nullity and Rank of Matrix

Given $A$ is an $m\times n$ matrix, where $m<n$. Since $m<n$ then I can conclude that the reduced form $A$ will have free variables, hence $dim$ $nullA>0$. Let's say $dim$ $nullA=1$. Since $m&...
1
vote
0answers
37 views

Rank of block matrix with equal diagonals

Let $$ Q=\begin{bmatrix}\phantom{-}A&B\\-B&A\end{bmatrix} $$ be a block matrix where $A,B\in \mathbb{R}^{n\times n}$. Prove that $$\mathrm{Rank}(Q)=2\mathrm{Rank}(\left[\,A \ \: B\,\right]).$$ ...
0
votes
1answer
24 views

$|\text{rank}A-\text{rank}B|\leq \text{rank}(A+B)$

I would like to prove the following inequalities: $|\text{rank}A-\text{rank}B|\leq \text{rank}(A+B)$ I know that $\text{rank}(A+B)\leq \text{rank}A + \text{rank} B,$ but I can't tackle the problem. ...
0
votes
0answers
34 views

Rank of two binary matrices

Let $N=2^n$. Consider two matrices $M,P$ over $GF(2)$ where $M$ is a circulant matrix of size $(N,N)$. Matrix $P$ is of size $(N,N+1)$. All values of $P$ are same as $M$ except last column. Also $P_{1,...
0
votes
0answers
9 views

Conditions under which the product of pre and post multiplying a diagonal matrix leads to an invertible matrix

For some $n \times n$ matrix, $A$, that is diagonal/invertible and any $n \times m$ matrix, $B$, is there any way to tell immediately if the following product: $B^TAB$ is invertible? It's clearly ...
-1
votes
1answer
24 views

How to prove $rank(A^+)$ is no more than rank(A)? [duplicate]

Look here, my friend. How to prove the following equation? Or give a counter-example. $$\text{rank}(A^+)\leq \text{rank}(A)$$ where $A$ has full rank, and $\text{rank}(A^+)$ represents the positive ...
1
vote
1answer
34 views

How to prove rank(A+) is no more than rank(A)? [closed]

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 ...
0
votes
0answers
18 views

How to get the partial information matrix from the covariance matrix

I have four random variables${\left( {x,y,z,\varphi} \right)^T}$. And I know its covariance matrix $Cov=\left[ {\begin{array}{*{20}{c}} {\sigma _{xx}^2}&{\sigma _{yx}^2}&{\sigma _{zx}^2}&{...
4
votes
1answer
44 views

Convex polygon inscribed in a circle with vertices that create matrix. Prove that the rank of this matrix is less or equal 2.

Consider a convex polygon inscribed in a circle with vertices $P_1$, ..., $P_n, \ n \ge 3$. Let $A$ be the matrix $n \times n$ such that $\begin{equation} a_{ij} = \begin{cases} |P_iP_j| ...
3
votes
3answers
80 views

Let $A,B\in\mathbb{C}^{n\times n}$ be such that $A^*B=B^*A$. Show that $\text{rank} (A,B)=n$ iff $\det(B+i A)\neq0$

Does anybody know how to prove the following statement: Let $A,B\in\mathbb{C}^{n\times n}$ be such that $A^*B=B^*A$. Show that $\det(B+iA)\neq0$ if and only if $\operatorname{rank} (A,B)=n$. It seems ...
2
votes
2answers
31 views

Is the rank of a matrix with coefficients $\{-1,0,1\}$ the same as the rank of the matrix with coefficients in $GF(3)$?

I have a set of matrices defined over the ring of the integers, which items are using only coefficients -1, 0 and 1. For example: $$ A = \left(\begin{matrix} 1 & 0 & -1 \\ -1 & 1 &...
0
votes
0answers
30 views

Rank of the element-wise square of a matrix

I am looking for a real matrix A of rank r with non-negative entries with the following property : For every complex matrix B such that $B\circ \overline B=A$, the rank of B is strictly greater than ...
2
votes
1answer
38 views

Can we perturb a low rank map to a full rank map in a smooth way?

Let $f:\mathbb{R}^n \to \mathbb{R}^n$ be smooth. Can we find, for every $\epsilon>0$, a $C^1$ map $\tilde f:\mathbb{R}^n \to \mathbb{R}^n$ of full rank such that $\|df-d\tilde f\|_{C^0}<\...
1
vote
2answers
56 views

Can $A + uv^T$ be non-singular if A(order n) has rank n-1

Suppose $A$ is a matrix of order $n$ with rank n-1, $u$ and $v$ are $n$-vectors. Can $A + uv^T$ be nonsingular, if yes then find such $u$ and $v$. First I wrote Rank$(A + uv^T) <=$ Rank$(A)$ + ...
0
votes
0answers
31 views

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 &...
0
votes
0answers
24 views

Rank of sum of two matrices with identity matrix

Suppose I have partitioned matrices as \begin{equation} A = \left( \begin{array} {c,c} 0_{m \times n} \quad I_{m}\end{array} \right) \end{equation} \begin{equation} B^{T} = \left( \begin{array} {c,...
0
votes
1answer
24 views

Rouché-Capelli theorem

I understand what the theorem is about: [ https://en.wikipedia.org/wiki/Rouch%C3%A9%E2%80%93Capelli_theorem ][1] i.e., the rank of the augmented matrix wrt the rank of the coeff matrix provides ...
2
votes
1answer
33 views

Connection between ranks of an endomorphism and its linear image on the exterior power

Let $V$ be an $n$-dimensional real vector space, and let $1<k<n$. Let $\psi:\text{End}(V) \to \text{End}(\bigwedge^kV)$ be the exterior power map, $\psi(A)=\bigwedge^k A$. For $B \in \text{End}...
0
votes
1answer
16 views

What is the angle between a $A_{3x3}$ with $Rank(A)=2$ and $A^T$

I need to solve this problem: For Matrix $A_{3x3}$ with $Rank(A)=2$. If Matrix A is Transposed and its elements are the same as elements of Matrix B. What is the angle of rotation from A to B?
2
votes
1answer
63 views

How far from a matrix the rank does not decrease?

Let $A$ be a real $n \times n$ matrix, and suppose that $\text{rank}(A)=k$. Consider $$ r_0=\sup \{r>0 \,\, | \,\, |B-A| \le r \Rightarrow \text{rank}(B) \ge k \}.$$ Does $r_0$ depend only on ...
0
votes
3answers
24 views

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
votes
1answer
24 views

Deleting any column from this matrix doesn't change the rank?

Suppose we have a $m\times n$ matrix $A$ such that $\text{rank}(A) = n-1$ and the nullspace of $A$ is 1-dimensional and contains the all-ones vector, $e$. That is, $Ae=0$. I'd like to show that if we ...
0
votes
1answer
16 views

Trouble moving forward in the proof of Rank Nullity theorem

I understood that for a $m*n$ matrix $A$, if its rank is $r$, $(n-r)$ pivot free columns correspond to $(n-r)$ free variables. But if thats the case, how do I go about proving that the dimension of ...
4
votes
2answers
101 views

Rank/determinant of the $n\times n$ matrix $((a_{ij}))$ where $a_{ij}=(i+j-1)^2$

I am trying to find the rank and determinant of the following $n\times n$ matrix : $$A=\begin{bmatrix}1^2&2^2&3^2&\cdots&n^2 \\ 2^2&3^2&4^2&\cdots&(n+1)^2 \\\...
0
votes
2answers
30 views

Rank, invertibility and solution spaces

Let A and B be two matrices of the same order. Let's assume that ρ(B)=ρ(AB) (ρ=Rank). Which of the following two statements are correct or incorrect? A is invertible The solution space of $AB\mathbf ...
4
votes
2answers
55 views

What is the expectation of the rank of a matrix with a 1 at each column?

Say a random square matrix $A\in\mathbb{R}^{n\times n}$, each column of $A$ has exactly one nonzero element being 1, i.e. each column looks like $e_i=\{0,\dots,1,\dots,0\}^\top$. Say for each column, ...
0
votes
0answers
17 views

Efficient way to iteratively compute span of vectors.

The problem itself is not difficult, but the naive approach seems very wasteful (computationally speaking). Given an ordered list of $n$-dimensional vectors $(v_1 ,\dots ,v_m)$, that span $\mathbb{R}^...
0
votes
2answers
51 views

How to prove this matrix is rank n?

Let $M = \begin{pmatrix} p_0 (1- p_0) & -p_0 p_1 &\ldots & p_0 p_n\\ -p_1 p_0 & p_1 (1-p_1) & \ldots & p_1 p_n\\ \vdots & & &\vdots\\ -p_np_0 &\ldots&&...
4
votes
0answers
84 views

Upper bound CP tensor rank

I have a question about CP tensor ranks. In the following, $\mathcal X \in \mathbb R^{n_1 \times n_2 \times n_3}$ is a third-order tensor of CP rank $R$, i.e., there exist vectors $a_i$, $b_i$ and $...
0
votes
1answer
30 views
1
vote
0answers
35 views

What is a rank-1 tensor? What is the meaning of rank in this context?

I feel like different sources use the term "rank" differently, which is perhaps leading to my confusion. When I think of rank I think of number of linearly independent columns/rows, number of pivots ...
0
votes
1answer
47 views

Rank(AB)=Rank(A) if and only if Null Space (A) ∩ Image(B)={0}

For any matrices A and B of conformable dimensions. I think I got the "if" part, but I'm stuck in the "only if". I tried a counterpositive argument but I failed to find a contradiction.
0
votes
0answers
13 views

Linearly Dependent Rows and Rank Graphical Understanding

I was having trouble today understanding the correlation between linearly dependent rows and the rank of a matrix. Based on my teaching, we can determine rank by solving for reduced row-echoleon form ...