Questions tagged [matrix-rank]

For questions regarding the rank of matrices in linear algebra.

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47 views

Doubt regarding the proof of row rank = column rank

Wikipedia provides two methods to prove row rank of a matrix is equal to its column rank. My doubt is regarding the second method. But the wikipedia page mentions that this proof is valid only for ...
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1answer
21 views

Accuracy of low rank approximation

I am currently studying about randomized low-Rank Approximation of a matrix. In the problem's statement, given $m$ x $n$ $A$,it is referred that we want to minimize $\|A-Q_{k}Q^{T}_{k}A\|$ and for ...
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1answer
48 views

What can you say about $T$ if dim$(V) =$ Rank$(T - \lambda I)$?

I stumbled across this condition and I wanted to know what you could say about this: Let $T:V \to V$ be a linear transformation, with $V$ having a finite dimension. What can you say about $T$ if ...
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1answer
39 views

Symmetric Matrix over a finite field of Characteristic 2

Let $M$ be a $n$ by $n$ symmetric matrix over a finite field of Characteristic 2. Suppose that the entries in the diagonal of $M$ are all zero, and $n$ is an odd number. I found that the rank of $M$ ...
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1answer
83 views

Rank of some power of a given matrix

$A$ is a $3\times 3$ real matrix such that rank of $A^3$ is $2$. What is rank of $A^6$? Since $A^3$ has rank $2$, $A$ also has rank $2$ because the determinant is zero. Can I decisively conclude that $...
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1answer
50 views

Eigenvalues and rank connection

Let $A$ be a $4 \times 4$ matrix and let $\dim \mathcal N (A) = 2$. What can you tell about the eigenvalues of $A$?
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1answer
20 views

Rank of a left-orthogonal decomposition $A^T=UB^T$

I've got a rather simple question, but couldn't find an answer to it: Say $A\in\mathbb R^{m\times n}$ can be decomposed according to $$A^T=UB^T\in\mathbb R^{n\times m}\tag1$$ for some left-orthogonal (...
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0answers
9 views

Convert a rank deficient matrix to a full rank matrix

Suppose that $A$ is an $n\times n$ positive semi-definite matrix of rank $n-k$ where $k\geq 1$. Assume that we know $k$ independent null vectors $v_1,\dots,v_k$ of $A$. Is there a way to transform $A$ ...
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2answers
24 views

How does the linear independence or dependence of the set of column vectors of a matrix depend on that of the set of row vectors of the same?

Let A = ($a_{ij}$) be an mxn matrix. If the set of row vectors of A is linearly independent, is the set column vectors too? What happens if the row vectors are linearly dependent. Does it affect the ...
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1answer
29 views

Different rank and nullity obtained from intuition and computation

I just started learning linear algebra in school. After watching 3Blue1Brown's Essence of Linear Algebra, and there's something I'm confused about. \begin{pmatrix} 2&0\\ -1&1\\ -2&1\\ \...
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0answers
39 views

Range constraints with a full rank matrix

I am trying to execute a quadratic program using cvxopt. My problem is a very simple issue related to matrix rank in the constraints. I want to enforce that some of my variables are in the range [0,1] ...
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1answer
43 views

Design a positive integral matrix that isn't full rank

Is it possible to design a square $d \times d$ matrix with value $2^n$ on the main diagonal ($n \geq 2$ not given) all off-diagonal elements chosen from the set $\{ 1, 2, 2^2, \dots, 2^{n-1} \}$ ...
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1answer
10 views

Pseudoinverse of block diagonal matrix

Suppose I have some block diagonal matrix $A$, defined as: $A = \begin{bmatrix} A_1 & 0 & ... & 0 \\ 0 & A_2 & ... & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 &...
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5answers
38 views

Why does adding $\lambda \boldsymbol{I}$ to $\boldsymbol{X}^T\boldsymbol{X}$ for $\lambda > 0$ guarantee invertibility?

This question is inspired by regularized least squares, where it is stated that $$ X^TX + \lambda I $$ is guaranteed to be invertible for all $\lambda > 0$. Is there an intuitive reason for how ...
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1answer
12 views

Lower Bound on Rank of a Boolean Matrix

Let $M$ be a Boolean Matrix, i.e. $M \in \{0,1\}^{n\times m}$. I want to figure out a tight lower bound for rank of the matrix $M$. Rank of a matrix is defined here(wikipedia) for your reference. In ...
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1answer
52 views

Some use this matrix to disprove that the number of eigenvalues is equal to rank.

Claim: For any matrix, the number of non-zero eigenvalues – including algebraic multiplicity – seems to always be equal to the rank of the matrix. Here is a non-counterexample: \begin{bmatrix} 3 &...
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2answers
80 views

Prove that a non-zero vector cannot belong to the rowspace and nullspace of a matrix at the same time.

With research, I've mainly found that rowspace is the orthogonal complement of the nullspace, and the only vector that belongs to both spaces at the same time is {0}. In the linear algebra subject ...
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1answer
32 views

Matrix-Matrix Multiplication Properties

Let $A,B$ be $n\times n$ matrices in $\mathbb{R}^n$ where $AB=I$. Prove that rank$B=n$. I have reviewed the Systems Rank Theorem. I am thinking I have to prove that B in linearly independent which ...
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1answer
21 views

Does equality hold in Frobenius inequality if $XZ=I$

The Frobenius inequality states that: $$\operatorname{rank}(XY)+\operatorname{rank}(YZ) \le \operatorname{rank}(Y)+\operatorname{rank}(XYZ)$$ My question: Does the equality hold in Frobenius ...
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1answer
30 views

Singularity of a Symmetric Complex Matrix

Let $B\in\mathbb{R}^{n\times n}$ be a real symmetric positive definite matrix. Note that since $B$ is positive definite, it is non-singular. Does there exist any real symmetric matrix $A\in\mathbb{R}^...
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1answer
30 views

Expressing rows in terms of other rows in a matrix

how do I explicitly prove that a matrix of this kind: $$A=\begin{pmatrix}1&1&0&0&0\\ \:1&1&0&0&0\\ \:0&0&0&0&0\\ \:0&0&0&0&0\\ \:0&...
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3answers
23 views

Sum of Ranks of Two Complementary Matrices

Suppose we have a binary matrix $A$, i.e, all the elements of $A$ are either $0$ or $1$. Let $B$ be the complementary matrix of $A$, i.e., $$ B_{ij}= \begin{cases} 1,\textrm{ if }A_{ij}=0;\\ 0,\...
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1answer
59 views

$\operatorname{rank}(A)=2$, $\operatorname{rank}(B)=1$ and $\operatorname{rank}(C)=2$. Find $\operatorname{rank}(ABC)$.

The question was that whether $\operatorname{rank}(ABC)$ is equal to $1$ or not. The matrices are $3\times3$. So I wanted someone to help me understand this question without using formulas but giving ...
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0answers
13 views

Need help to explain the objective function with numerical example.

ai = element in i position in matrix? Cj= column j? Xc(i,j) = X in cluster following (i,j) position in matrix?
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3answers
137 views

Find the rank of $T^2$

Question: Let $\mathbb{C}^{11}$ is a vector space over $\mathbb{C}$ and $T:\mathbb{C}^{11}\to \mathbb{C}^{11}$ is a linear transformation. If dimension of Kernel $T=4$, dimension of Kernel $T^3=9$ and ...
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1answer
20 views

Solving Ax = 0 , pivot variables, free varaibles, need more specific explanation

Recently i started to lear linear algebra out of MIT OpenCourceWare. But i cant quite understand one little 'proof' (demontration) of getting $N(A)$ out of expression $Ax = 0$ by creating rref form of ...
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1answer
21 views

Is the identity $dim(col(AB))=dim(row(A) \cap col(B))$ correct?

I want to find an identity of dim(col(AB)) to gain some insight into the behavior of rank during matrix multiplication. For an $m \times n$ matrix A and an $n \times p$ matrix B, is the identity $dim(...
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1answer
31 views

The continuity of ordered eigenvalues of a matrix with function elements

Let $A$ be a matrix of $n\times n$ whose element is continous functiions on $\mathbb{R}^n$. Assume $A$ is an Hermitian matrix at every point of $\mathbb{R}^n$, which means all its $n$ eigenvalues are ...
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2answers
36 views

What is a solution to this matrix rank problem?

Find the rank of a matrix depending on parameters r, s. \begin{bmatrix} 1 &0 &0 \\ 2 &r-2 &2 \\ 0 &s-1 &r+2 \\ 0 &0 &3 \end{bmatrix} My attempt was to ...
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1answer
18 views

Let $A,B\in M_{10}(\Bbb{R})$ such that $A,B$ have rank $3,2$ respectively. If Img$B\subset$Coimg$A$,then find rank of $AB$.

By definition $\operatorname{Img}B$ denotes the column space of $B$ i.e. $\operatorname{Img}B=\{Bx\in\Bbb{R}^{10}\ |\ x\in\Bbb{R}^{10}\}$ and $\operatorname{Coimg}A$ denotes the row space of $A$ or ...
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0answers
18 views

Is there any best numerical iteration methods than perform faster than reduced row echelon method?

In order to find the best low-rank approximation, are there any numerical methods perform faster than using a reduced row echelon form to find the rank of a matrix?
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17 views

Two rectangular matrices are equivalent matrices if and only if they have the same rank

We will call two rectangular matrices $A$ and $B$ of the same dimensions equivalent if there exist two non-singular matrices $P$ and $Q$ such that $B = PAQ$ prove that $A$ and $B$ are equivalent if ...
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0answers
23 views

Singular behavior of a system of linear equations

I have the system of 6 linear equations for 9 variables. It has the three-dimensional space of solutions, and everything looks fine except the case $\theta=0$. In this case some variables go to ...
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2answers
66 views

$X^TX$ not full rank when $X$ is full rank?

$X$ is an $n \times m$ matrix, where $n \geq m$ and $\mbox{rank}(X) = m$. Is it possible for $X^TX$ to not be full rank? If $X$ can only be square I could easily prove this with $$ \det(X^TX) = \...
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0answers
28 views

Which variable cannot be zero in underdetermined system?

Consider the underdetermined system $Ax=b$, where $A \in R^{m\times n}$ with $m<n$. If the Matrix $A$ has rank $m$, my question is which variable of $x_1,x_2,\dots,x_n$ cannot be zero? I mean if ...
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1answer
33 views

Find $m$ so that $r(A^{-1})=2$.

Let $A=\begin{pmatrix} 1 &3 &1 &1 \\ 2 &2 &-1 &0 \\ 2 &4 &-2 &0 \\ 0 &-1 &2 &m \end{pmatrix}$ Find $m$ so that $r(A^{-1})=2$. I have ...
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2answers
53 views

Why doesn't Gram-Schmidt work for all matrices?

So I'm doing a MATLAB assignment where the objective is to write code to do the Gram Schmidt algorithm. I just want to say, my code is almost certainly not the issue, as the same issue arises in ...
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1answer
36 views

Find $m$ so that $\mbox{rank} \left( A^{-1} \right) = 3$ [closed]

Let matrix $$A=\begin{bmatrix} 1 & 0 & 2 \\ 2 & 3 & m \\ 3 & 4 & 2\end{bmatrix}$$ Find $m$ so that $\text{rank} \left(A^{-1}\right)=3$ Please help me with this problem. ...
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0answers
15 views

The necessary and sufficient condition of $rank(A)+rank(B)=rank(A+B)$ [duplicate]

Given two $m \times n$ matrices, find the necessary and sufficient condition of $rank(A)+rank(B)=rank(A+B)$. Here is my idea: Intuitively, I think it is like that A and B compliments each other. ...
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0answers
36 views

Nonlinear least squares uniqueness

Suppose I have a nonlinear least squares objective function I want to minimize: $$ \chi^2(\mathbf{x}) = \sum_{i=1}^n f_i(\mathbf{x})^2 = \mathbf{f}(\mathbf{x})^T \mathbf{f}(\mathbf{x}) $$ Now suppose ...
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1answer
20 views

What will the rank be, if a matrix multiply its transpose?

Let's say we got a matrix $A_{n\times k}$ and $n>k$, and the column rank of $A$ is $k-1$ if we pre- and postmultiply its transpose, i.e. $A^TA$ and $AA^T$. Is the rank of $A^TA$ and $AA^T$ still $...
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1answer
16 views

A question from TIFR GS related to rank of a matrix.

I found this problem in TIFR-GS paper.I have also solved this problem.Can someone please tell me if there are more interesting facts hidden in this problem that needs my attention or which I have ...
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1answer
20 views

'For transformation matrices, the rank of matrix tells you the dimensions of the output.' What does this statement mean?

Kindly explain in simple terms because I have limited knowledge of matrices. Also, what is a transformation matrix?
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1answer
46 views

If $A,B,C\in M_n(\mathbb{R})$ and $A+B+C=0$, then what are the possible values of the triple $\big(\mbox{rank}(A),\mbox{rank}(B),\mbox{rank}(C)\big)$?

If $A,B,C\in M_n(\mathbb{R})$ and $A+B+C=0$, then what are the possible values of the triple $\big(\mbox{rank}(A),\mbox{rank}(B),\mbox{rank}(C)\big)$? I know that $\mbox{rank}(A)\leq \mbox{rank}(B)+\...
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1answer
36 views

If all minors are $0$, then rank is at most $n-2$ [duplicate]

Can anyone prove/disprove the following statement? Given a square matrix of size $n\ge2$, if all of its $(n-1)$-rowed minors are zero, then $\operatorname{rank}(A)\leq n-2$. I'm having trouble ...
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2answers
24 views

rank of block matrix whose diagonal blocks are invertible

Suppose I have a block matrix $$P = \begin{bmatrix} A & B \\ C & D\end{bmatrix},$$ where $A\in\mathbb{R}^{n\times n}$ and $D\in\mathbb{R}^{m\times m}$ are invertible. $B\in\mathbb{R}^{n\times ...
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2answers
77 views

If all minors are $0$, the rank is at most $n-2$

Can anyone prove/disprove the following statement? Given a square matrix of size $n\ge2$, if all of its $(n-1)$-rowed minors are zero, then $\operatorname{rank}(A)\leq n-2$. I'm having trouble ...
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2answers
80 views

Square Matrix Inequality

Suppose that for two $n \times n$ matrices $A,B$, $AB = A + B$. Prove that $$\text{rank}(A^2) + \text{rank} (B^2) \leq 2 \text{rank} (AB).$$ This reminds me of Sylvester's Rank Inequality theorem, ...
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0answers
18 views

Vandermonde and Cauchy matrix

I got a question regarding a type of matrix that I got from this paper, "A Tutorial on Reed-Solomon Coding for Fault-Tolerance in RAID-like Systems". There is a matrix for the reed-solomon encoding: ...
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3answers
38 views

Prove $ \dim L(\{v_1+w,v_2+w,\dots,v_n+w\})\geq n-1,w\in V$

Let $v_1,v_2,\dots,v_n\in V, $ linearly independent Prove $ \dim L(\{v_1+w,v_2+w,\dots,v_n+w\})\geq n-1,w\in V$ I tried $\lambda_1(v_1+w)+\lambda_2(v_2+w)+\dots+\lambda_n(v_n+w)=0$ $\lambda_1v_1+\...

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