Questions tagged [matrix-rank]

For questions regarding the rank of matrices in linear algebra.

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If $A$ and $B$ are normal with $\sigma(A),\sigma(B)\subseteq \mathbb{R}\cup\mathbb{T}$, does $\text{dim ker}(AB-BA)=\text{dim ker}(A^*B-BA^*)$?

I already asked this question for general $B$ and it was answered negatively here, so if the statement is true, one has to use the assumption on $B$ as well. In the original question I already ...
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Finding rank of matrix $A$.

Let $A_{n×n}$ =$((a_{ij}))$ n≥3 , where $a_{ij}=(b_i^2−b_j^2)$ , i,j=1,2,…,n for some distinct real numbers $b_1,b_2,…,b_n$ . Then what is $\rho (A)$ ? My attempt: A can be written as difference of ...
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Proving that the dimension of the kernel of this map is $1$

Let $0 \neq z \in \mathbb{F}^n_2$, and define $A_z = \{ y\in \mathbb{F}^n_2 \mid z \cdot y = 0\}$, where $\,\cdot\,$ is the dot product $$ z \cdot y = y_1 z_1 + \dots + y_n z_n \pmod 2. $$ Suppose $...
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If the trace of a matrix equals its rank, is it idempotent?

It is well-known and can easily be proven that if a matrix $A$ is idempotent, then its trace equals its rank: $$ A^2 = A \Rightarrow \mathrm{tr}(A) = \mathrm{rk}(A) $$ Does the inverse also hold? If ...
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$\mathrm{rank}(A^k)=\mathrm{rank}(A^{k+1})$ for some $k \ge 1$ implies $\mathrm{rank}(A^m)=\mathrm{rank}(A^k)$ for every $m \ge k$ [duplicate]

During my studying linear algebra, I came across the following problem: Let $A$ be a square matrix with real entries, $k$ be a positive integer, and assume $\operatorname{rank}(A^k)=\operatorname{...
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Rank and Nullity of Matrix with Arbitrary Parameters

Given the matrix A defined as A = \begin{bmatrix} 2 & -4 & 4 & -2 \\ 6 & a-2 & 7 & a-6 \\ -1 & 2-b & 8 & -b+1 \\ \end{bmatrix} where (a) and (b) are real constants, ...
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If $A$ is normal with $\sigma(A)\subseteq \mathbb{R}\cup\mathbb{T}$, does $\text{dim ker}(AB-BA)=\text{dim ker}(A^*B-BA^*)$?

This clearly holds if $A$ is self-adjoint, and also if $A$ is unitary, because then $A(\text{ker}(AB-BA))=\text{ker}(A^*B-BA^*)$. To prove this, if $w\in\text{ker}(AB-BA)$, then $A^*B(Aw)=A^*ABw=Bw=BA^...
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computational complexity of stabilization problem in Boolean control network

Stabilization problem is a fundamental problem in control theory. There are many literature to achieve stabilization but fewer results are related in computational complexity. Consequently, we ...
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Generalization of Sylvester's law of inertia to the case of rectangle matrix

Sylvester's law of inertia states given a symmetric matrix $A$ and a squared invertible matrix $S$ of the same size, then $A$ and $SAS^\top$ have the same number of positive, negative, and zero ...
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Linear algebra and inequality

I have the following 6 inequalities: $$ x_1 + x_2 + x_4 \le K \\ x_3 + x_4 + x_5 \le K \\ x_1 + x_5 + x_6 \le K \\ x_1 + x_2 + x_6 \le K \\ x_2 + x_3 + x_4 \le K \\ x_3 + x_5 + x_6 \le K $$ where $x_1$...
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How to prove $\tilde{A}=O$ if $\mathrm{rank}(A) \le n - 2$

In the final exam of the linear algebra course I took, the following problem was asked and I wasn't able to solve it: Let $n \ge 2$ and $A$ be a $n \times n$ matrix. If $\mathrm{rank}(A) \le n - 2$, ...
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Having trouble with econometrics VECM question

Hi im having trouble solving this mcq question on econometrics. Could someone help me? Thank you! Full question here
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Finding Dimension of Preimage of Subspace under Surjective Linear Transformation: A Linear Algebra Problem

"I am currently working on solving a previous year Linear Algebra exam question. The question involves vector spaces $V_1$ and $V_2$ over the same field $F$, with a surjective linear ...
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If $A$ is normal and $A$ and $B$ almost commute, do $A^*$ and $B$ almost commute?

Suppose, $A,B\in \mathbb{C}^{d\times d}$, $A$ is normal, and $\text{dim ker}(AB-BA)\ge (1-\delta)d$. Does then follow that $\text{dim ker}(A^*B-BA^*)\ge (1-\delta)d$, possibly for different $\delta$? ...
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Matrices with rank 1 over finite field

Let $\mathbb{F}_{11}=\{\overline{0},\overline{1},\ldots,\overline{10}\} $ be a finite field and $A=(a_{ij})_{3 \times 3}$ a matrix such that $a_{ij} \in \mathbb{F}_{11}, \ \ \forall \ 1\leq i,j\leq 3$...
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matrix with univariate entries: rank deficit of specialization ≤ vanishing order of determinant, part II: commutative ring

a special case of this question with coefficients in a field was recently asked and answered. fix a commutative ring with unit $R$, and an $n \times n$ matrix $M(X)$ with entries in $R[X]$. the ...
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matrix with univariate entries: rank deficit of specialization ≤ vanishing order of determinant.

fix a field $F$ and consider an $n \times n$ matrix $M(X)$ with entries in $F[X]$. the determinant $\det(M(X))$ is itself a polynomial, say $D(X)$. clearly, if $x \in F$ is such that the specialized $\...
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Can anyone help with this matrix problem? [duplicate]

enter image description here In this image, there's a question about asking to get the determinant of matrix D. I'm stuck on how to represent it as using only a and n.
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Specific basis for the space of symmetric matrices

Consider the space of symmetric matrices $symm(M)$ over reals of dimension $n \times n$. It is clear that there is a straightforward basis for this space where for any $i \ge j$ $M_{ij}(m,n) = 1$ if $...
supernova's user avatar
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Determine all matrices, $A$, that satisfy the condition that $\operatorname{rank}(A^k) = \operatorname{rank}(A)$ for each $k \geq 1$.

My intution is that the condition is true if and only if $A$ is invertible or $A$ is a projection, i.e $A^2 = A$. It is trivial to show that if $A$ is invertible or $A$ is a projection then the ...
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If $A^TA$ is invertible and $W$ is square what is $\operatorname{rank}A^TWA$? [closed]

Consider $A ∈ \mathbb{R}^{m \times n}$ and $W ∈ \mathbb{R}^{m \times m}$ such that $n < m$ and $A^TA$ is invertible. If we know that $ \operatorname{rank}W = w \ge n $ what can we say about $\...
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Prove $\ker(A+T(A))\subseteq \ker(A)$ for $A\geq 0$ and $T$ positive linear map

Let $A\geq 0$ be a positive semi-definite complex matrix in $M_d(\mathbb{C})$. Let $T:M_d(\mathbb{C})\to M_d(\mathbb{C})$ be a positive linear map between $d\times d$ complex matrices, i.e., $A\geq 0\...
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Finding the entries in a matrix given that the rank has to be $2$

Find all $y<0$ such that the matrix defined for $y<0$ $$\begin{pmatrix}-3&2&y\\ 0&1&-\frac{1}{y}\\ y&0&y\end{pmatrix}$$ has rank $2$. It seems ok, we just need a $0$ row, ...
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Are the columnspace and the nullspace not always the same for some matrix A (mxn) and some product of matrices BA, where B is invertible (mxm)?

Why is this not always true? My rationale is that some vector c (mx1) must either be in the columnspace or nullspace of A. If c is in the nullspace of A, it must be in the nullspace of BA. If c is the ...
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Computing the rank of a $3\times 3$ matrix with some restrictions

I have the following system of equations: $$ 0 = \left[ (p_{31} - p_{11})^2 - \omega_{22} (p_{21} - p_{11})^2 \right] x^2_1 + \left[(p_{32} -p_{12} )^2 - \omega_{22} (p_{22} - p_{12})^2 \right] x_2^2+ ...
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Intuition behind pre-multiplication of a vector with a matrix

Given a matrix $\mathbf{A}$ and a vector $\mathbf{x}$, the MVP $\mathbf{Ax}$ gives the geometric interpretation of transforming $\mathbf{x}$ into the coordinate system with basis vectors defined by ...
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Implication of a certain matrix being full row rank regarding the rank of another matrix

Consider the matrices $$X = \begin{bmatrix} x_0 & x_1 & ... & x_{N-1} \end{bmatrix} \in \mathbb R^{n\times N}\\ U = \begin{bmatrix} u_0 & u_1 & ... & u_{N-1} \end{bmatrix} \in \...
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Decide whether a 0-1 matrix has full rank

Is there a theorem about when a (sparse) 0-1 matrix $A = (a_{ij})_{\substack{1\leq i\leq n\\ 1\leq j\leq m}}$ has full rank? More concretely, I know the row sums of $A$, $$ r_i = \sum\limits_{j=1}^m ...
mathaholic's user avatar
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Does $M = \sum_{k} \mathbf u_k \mathbf v_k^{\top}$ imply $\mathbf u_k \in \operatorname{colspan}(M)$?

Let $M = \sum_{k} \mathbf u_k \mathbf v_k^{\top}$, where both sets of vectors $\{\mathbf u_k\}$, $\{\mathbf v_k\}$ are linearly independent. Can we conclude that $\mathbf u_k \in \operatorname{colspan}...
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$rank\begin{pmatrix}f(A)\\g(A)\end{pmatrix}=rank\begin{pmatrix}f(A)&g(A)\end{pmatrix}$?

Let $A$ be an $n\times n$ matrix. For any polynomials $f,g$, show that $$rank\begin{pmatrix}f(A)\\g(A)\end{pmatrix}=rank\begin{pmatrix}f(A)&g(A)\end{pmatrix}.$$ I just could say $$rank\begin{...
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How to find a matrix perturbation which lowers the rank of a matrix

I have a matrix $A \in \mathbb{R}^{m x n}$ which has independent columns. I want to find the smallest perturbation which will make it have a kernel and a vector in that kernel. Something like $$ \min_{...
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Linear independence of binary vectors

I lately encountered the following problem, which I conjecture to be true, but have been unable to either give a proof or raise a counterexample. Let $A$ be a $m \times n$ binary matrix ($m \ge 2n, n \...
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Let A be an m by n matrix and let B be an n by p matrix. Show that rank A + nullity A = n is the special case of rank AB + dim(null A ∩ col B)=rank B.

Let $A$ be an $m \times n$ matrix and let $B$ be an $n \times p$ matrix. Show that $\text{rank } A + \text{nullity } A = n$ is the special case of $\text{rank } AB + \dim(\text{null } A \cap \text{col ...
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Solvability of $A X B=C$ with $X=X^\mathrm{T}$

I am studying symmetric solutions to the complex matrix equation \begin{equation} A X B=C, \end{equation} where $A$, $B$, and $C$ are $m\times n$, $n \times k$, and $m \times k$ complex matrices, ...
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How is effective rank (a ratio of nuclear norm and operator norm) defined in terms of eigenvalues?

I am reading this paper on implicit regularisation in gradient descent and I am having difficulty with the provided definition of effective rank. In the paper it is given as $r(W) = \frac{||W||_*}{||W|...
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rank of block lower triangular matrix [duplicate]

Let us consider \begin{equation} D= \begin{bmatrix} A & 0 \\ B & C \end{bmatrix} \end{equation} where $Ax \neq 0$ with $x \neq 0$ and the number of rows of $A$ can be larger than that of ...
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row rank = column rank over a PID?

Suppose I have some matrix $M$ with entries from a PID. Define the column rank and row rank as you would expect: the rank of the module generated by the columns (resp. rows). Is it always true that ...
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Rank 1 matrix with given anti-diagonal sums

I'm looking for a matrix that is of rank 1 but has specified anti-diagonal sums. To give a few examples: It is easy to make a rank 3 matrix which has anti-diagonals that sum to 1: \begin{bmatrix} 1 &...
Håvard Arnestad's user avatar
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If $\begin{bmatrix}XY \\ X\end{bmatrix}$ has full column rank, then does $X$ have full column rank?

Consider the rectangular matrix $X \in \mathbb R^{m \times n}$, the square matrix $Y \in \mathbb R^{n \times n}$, and the block matrix $Z = \begin{bmatrix}XY \\ X\end{bmatrix} \in \mathbb R^{2m \times ...
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From adjacency Matrix can we find the maximum number of disjoint matching pairs of a simple undirected graph?

I came across this problem which boiled down to finding the maximum number of pairs of disjoint edges given a simple undirected graph. After doing some research I came across Edmond's Blossom ...
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Prove that if $R(T)\cap N(T)=\{0\}$ then $\operatorname{rank}(T)=\operatorname{rank}(T^2)$

In the Linear Transformations chapter in the book "Linear Algebra" by Michael O'Nan, there's this question : $R(T)\cap N(T)=\{0\}$ iff $\operatorname{...
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Determine the rank of this matrix manually.

I am trying to determine the rank of this matrix manually using the instructions found here: https://www.sangakoo.com/en/unit/rank-of-a-matrix-gaussian-method The matrix: ...
The Radiant's user avatar
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Is the set of vectors $(1, 0), (i, 0)$ a linearly independent subset of $\mathbb{C}^2(\mathbb{R})$?

I've encountered a problem with two approaches, leading to contradictory results. Approach 1: Utilizing the definition of linear independence, consider the equation $c_1 \begin{bmatrix} 1 \\ 0 \end{...
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Sum of symmetric matrices of rank one

We let $A$ and $B$ be symmetric matrices of rank one, where $A \neq kB$ for any $k$. We wish to show that the rank of $A + B$ is 2. By subadditivity, we have that the rank of $A+B$ is at most 2. For ...
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Existance of matrices such that their left null space contains another matrix with given dimensions

Let $A\in\mathbb R^{n\times c}, n>c$. Let $\operatorname{rank}(A)\leq c$. Hence the dimensions of the left null space of $A$ should be at least $n-c$. Given $B\in\mathbb R^{n\times d}$, if $d\leq n-...
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Effect of centering rows and columns on the rank of a full rank matrix

Let $A$ be a positive-definite and full rank matrix of size $n$. And let $H = I - n^{-1}1_n$ where $1_n$ is the square matrix of size $n$ full of $1$'s. We then build the centered version of $A$ (in ...
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Closed form error of inner-product-based low-rank approximation of binary matrix

Given a binary square matrix $A \in \{0, 1\}^{n \times n}$ and $1 \leq k \leq n$, we aim to compute the error of outer-product-based low-rank approximation. Formally, we aim to compute $$ \min_{X \in \...
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Rank of a matrix with identical diagonal, upper triangle, and lower triangle

I'm trying to figure out when the matrix below is full rank or non-singular. All the elements in the upper triangle (non-diagonal) are equal to $a$, elements in the lower triangle are equal to $c$, ...
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Inequality about rank of matrix

$A,B$ are $n\times n $ matrix over a field, and $AB=BA$. Let $C=(A\mid B)$ (an $n\times 2n $ matrix). How to prove the following inequality about rank. $$\text{rank}(A)+\text{rank}(B)\ge \text{rank}(C)...
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(Potentially) A Vandermode Matrix Question

I am interested in solving the following: Let $\lambda_1,\dots,\lambda_r$ be distinct, nonzero complex numbers. Prove that the matrix $$L := \begin{bmatrix} \lambda_1 & \lambda_2 & \dots & ...
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