# Questions tagged [matrix-rank]

For questions regarding the rank of matrices in linear algebra.

1,673 questions
Filter by
Sorted by
Tagged with
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 ...
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 ...
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 ...
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 $M$ ...
83 views

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 ...
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 ...
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 &...
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 ...
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 ...
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 ...
30 views

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 ...
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 ...
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 ...
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?
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 ...
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 ...
66 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+\...