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Questions tagged [matrix-rank]

For questions regarding the rank of matrices in linear algebra.

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Probability for a boolean matrix from a certain class to have full rank

While considering a certain type of computational problems, I have encountered the following probabilistic problem over the two element field $GF_{2}$. For $c\in \mathbb{Q}_{>0}$, let $p_{c}(n)$ ...
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Finding rank/trace for projection involving pseudo-inverse?

Define the projection matrix $H = X (X^T X)^- X^T$, where $(X^T X)^-$ is a pseudo/generalised inverse (ie. $A A^- A = A$). How can I find the rank of $H$? I know the following two facts: $H$ is ...
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$\DeclareMathOperator{\rank}{rank}\rank(A) = \rank(B)$, Prove there exist $U, V$ invertible matrices such that: $A = UBV$

$\DeclareMathOperator{\rank}{rank}$$\DeclareMathOperator{\Mat}{Mat}$Given two matrices $A, B \in \Mat_{m \times n}$ , as $\rank(A) = \rank(B)$. Prove there exist two invertible matrices: $$U \in \...
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Persistence of excitation and rank conditions

Assume to have a linear system of order $n$ $$ x(k+1) = Ax(k) + Bu(k) $$ which is minimal (reachable and observable) and the following trajectories $$ U=[u(1), ...,u(T)] \quad X=[x(1), ..., x(T)]. $$ ...
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Rank of an integer matrix modulo a prime

Suppose I have a matrix A with integer elements and a matrix A mod p, where p is a prime number. In what circumstances rank(A)=rank(A mod p)? Based on my understanding, 1) when det(A) is nonzero but ...
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Column Space, Rank and Matrix Concatenation

I have the following question: Given Matrices $A$ and $B$, the following relation exists between their column spaces: $$\text{col}(B) \subseteq \text{col}(A)$$ Then, which of the following is true ...
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How can I find rank of $A=\sum_{i=1}^4 x_ix_i^T$ without actually finding the matrix $A$?

Suppose $A=\sum\limits_{i=1}^4 x_ix_i^T$ where $x_1=(1,-1,1,0)^T,x_2=(1,1,0,1)^T,x_3=(1,3,1,0)^T$ and $x_4=(1,1,1,0)^T$. How can I find the rank of $A$ without explicitly finding the matrix $A$ ...
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invertibility of the block Vandermonde matrix

Consider a block Vandermonde square matrix $$X = \begin{bmatrix} \mathbf{I} & \mathbf{\Omega}(i_0) & \mathbf{\Omega}(2i_0) & \ldots & \mathbf{\Omega}((2p-1)i_0) \\ \mathbf{I} & \...
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if $A \in C^{2015,2015}$ and $rank(A) < 1000$ proof that $\dim(\ker(A+A^T)) > 15$

I want to solve that thesis: if $A \in C^{2015,2015}$ and $rank(A) < 1000$ proof that $\dim(\ker(A+A^T)) > 15 $ from the fact that $$\dim(im(A)) = \dim(im(A^T))$$ and $$ \dim(\ker(A))+\dim(im(...
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Comparing the ranks of leading princial minors of a square symmetric matrix

Let $A$ be a $n$ x $n$ real symmetric matrix and let $A_k$ denotes the $k$-th order leading principal minor matrix of $A$. Prove that for $0 \leq k \leq n-1$: $$Rank(A_{k+1})\leq Rank(A_k)+2$$ ...
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Rank-one update of eigenvalues

Let $\lambda_1,\dots,\lambda_n$ be the eigenvalues of the square and symmetric $n\times n$ real matrix $A$. Consider a rank-one perturbation $B=A+uu^T$, where $u$ is a real $n$-vector. Is there an ...
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Compute the rank of the matrix

If positive integers $p>q\ge2$, $p$, then how to prove the rank of the matrix $$ M= \begin{bmatrix} \underbrace{1\dots1}_p & -2p & \underbrace{1\dots1}_p \\ \underbrace{1\dots1}_q & -2q ...
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Projection Matrixes, $A,B$ such that $\text{Id} - (A+B)$ is invertible

This question was on an old qual exam and I have been stuck on it: Let $A,B$ be two real $5x5$ matrices such that $A^2=A , B^2 = B$ and $\text{Id} - (A+B)$ is invertible. Show Rank($A$)=Rank($B$). ...
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Dimension of nullspace and number of rows

A matrix $A$ has $10$ columns and dim(Null($A^{T}$ ))$=7$. The smallest possible number of rows of $A$ is $(A)$ $5$ $(B)$ $6$ $(C)$ $7$ $(D)$ $8$ $(E)$ $9$ I know that dim(...
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Rank, nullity and the number of rows of a matrix

I have this question here: Let $A$ be a matrix with $10$ columns, $dim$ $Null(A)=5$ and $dim$ $Null(A^T)=3$. How many rows does $A$ have? $a)$ $8$ $b)$ $3$ $c)$ $5$ $d)$ $10$ ...
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What is the meaning of Rank[A | b]? (Linear Algebra)

I got a question in my textbook where I am supposed to find if the linear system Ax=b is consistent. Then we are given some information. I do think I know how to solve this kind of problem but they ...
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In this link with ques $rank(AA^t)=rank(A)$ ,I don't understand.

In the link of this question there is a answer with 76 votes in that answer by which step we can conclude that I it will not work for field of complex numbers or a field with non-zero ...
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Given this 4x4 matrix, is there an easier way to calculate the eigenvalues?

This came up as a textbook question: Find the rank and 4 eigenvalues of A, where A is the 4x4 matrix with all 1 entries 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I am ...
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If we know the rank of a matrix r, can we assume that will have precisely r non-zero eigenvalues?

I have looked at many answers on the internet regarding the relationship between rank and eigenvalues, and all of them contain complex calculations and descriptions too advanced for me, a beginner ...
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Is not full rank matrix invertible?

Problem $A$ is a $4 \times 4$ matrix. It is known that $\text{rank}(A)=3$. Is matrix A invertible ? Attempt to solve $\text{rank(A)}=3 \implies \det(A)=0$ which implies matrix is $\textbf{not}$ ...
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Compute $\text{rk}(p)$ using Gauss reduction on $A$. Compute $\dim\big(\ker(p)\big)$.

Let $B = (1, X, X^2)$ be a basis for $\mathbb{R}_2[X]$ and $p ∈ \mathcal{L}\big(\mathbb{R}_2[X]\big)$ be the linear map defined by $p(1) = \frac{1}{3}(2 − X − X^2)$, $p(X) = \frac{1}{3}(−1 + 2X − X^2)$...
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constrained rank approximation

I'm trying to solve a problem similar to this problem. Instead of requiring the diagonals to be 0, I'd like to require columns of the low rank approximation to decrease in value while going down the ...
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Differing rank between non-homogeneous system and coefficient matrix

Assume that for a non-homogenous system of m linear equations in n unknowns, the rank of the coefficient matrix is less than the rank of the augmented matrix. What conclusion can necessarily be drawn? ...
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Rank, nullity and consistency for two matrices

I have this question here which says the following. Let $A$,$B$ be $3 \times 6$ matrices with the following properties. $(i)$ For every b$\epsilon \mathbb{R}^3$, rank$(A)$ $=$ rank$([A|b])$ $(ii)$ ...
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If $A$ is a matrix $10$ by $12$ and $A x=b$ is solvable for every $b$, then th column space of $A$ is?

If $A$ is a matrix $10$ by $12$ and $A x=b$ is solvable for every $b$, then th column space of $A$ is? The column space of $A$ is the whole of $R^m$ which is $12$ is this answer the right answer ?
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Assuming matrices $A,B,C \;\text{and}\;D\;$ are $ n\times n,$ show two ways that $(A+B)(C+D)=AC+AD+BC+BD$

My way first of showing this is by letting $A,B,C \;\text{and}\;D\;$ equal \begin{bmatrix} a_{ij} \end{bmatrix} \begin{bmatrix} b_{ij} \end{bmatrix} \begin{bmatrix} c_{ij} \end{...
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Finding the Upper Bound of the difference between the Inverse of the 2 matrix

Given that $ K = A^{-1} - B^{-1} + A^{-1} - A^{-1}BA^{-1}$, we need to find the upper bound of $K$ where matrix $A = C + I\rho$ and $ B = C_{x} + I\rho $ has dimension $n\times n$, $ C = RDR^{T}$ ...
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Find a example of $A$ be $4 \times 4$ matrix such that $A$ has rank $2$ but $A^2 =0 $?

Find a example of $A$ be $4 \times 4$ matrix such that $A$ has rank $2$ but $A^2 =0 $? My attempt : $$A=\begin{bmatrix} 0 & 0 & 1 &0\\0 & 0 & 1 & 0\\0 &0 &...
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Prove that $r(A)=\operatorname{tr}(A^2)$

Let $A\in M_n(\mathbb{C})$. Show that if $A^3=A$, then $r(A)=\operatorname{tr}(A^2)$. Since $A^3=A$, the possible eigenvalues are $0,1,-1$. I don't know from here how to compute the rank of $A$. ...
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When does $AB$ have linearly independent columns, if $A$ and $B$ are non-square matrices?

If $A$ is $m \times n$ ($m<n$), and its rows are independent $B$ is $n \times p$ ($p<n$), and its columns are independent We also know $m\ge n$. does $AB$ have linearly independent columns? ...
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Find the Jordan forms of a matrix from just the ranks of its eigenspaces

Let $C$ be a $10\times 10$ matrix whose characteristic polynomial is $(t+2)^5(t-3)^5$. Suppose that $rank((C+2I)^2)=6$ and $rank((C-3I))=8$. What are the possible Jordan forms of $C$? This is the ...
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Does the decay rate of singular values reflect linear dependency of vectors in a matrix?

In order to compare two matrices, suppose A and B, based on the level of linear dependency between column vectors of a given matrix, I can think about following measures: Rank of a matrix: If the ...
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Inverse of symmetric matrix $AA^\top$ where $rank(A)=m \leq n$?

Suppose I have a matrix $A\in\mathbb{R^{m\times n}}$ where $m\leq n$ and $rank(A)=m$. Is the matrix $AA^\top$ singular? My hunch is that the matrix is only non-singular when $n=m$.
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Rank minimization of a block matrix

When is the matrix $A = \begin{bmatrix}B & C \\X & D \end{bmatrix}$ have the minimal rank for given matrices B,C, and D? (matrices B, C, X and D have the size of $m\times n$, $m\times k$, $l\...
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Quadratic Forms rank and signature confusion

I would just like to clear some things up! A quadratic form can always be expressed in terms of a symmetric matrix. When we diagonalise this matrix we can read the rank and signature of the quadratic ...
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Find a response vector that minimizes the distance to the true line

I have a system of equation of the following form: $$ {\bf W} \vec{x} = \vec{y} $$ W is a thin matrix (i.e. has more rows than columns) and $\vec{x}$, $\vec{y}$ are vectors of appropriate size. I ...
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Repeated roots of $\det(A-xI)$

The solutions of the equation $\det(A-xI)=0$ give rise to the eigenvalues of Matrix $A$. This confuses me because the equation can give repeated roots. Suppose there is a root $x=\lambda$ of ...
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$\operatorname{rank}(A^2)+\operatorname{rank}(B^2)\geq2\operatorname{rank}(AB)$ whenever $AB=BA$?

Let $A,B$ be $n\times n$ matrices. If $AB=BA$, then $\operatorname{rank}(A^2)+\operatorname{rank}(B^2)\geq2\operatorname{rank}(AB)$. Is this rank inequality correct? No counterexample seems to exist. ...
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Prove That $T(B)=AB-BA$ has $\dim \operatorname{Im}(T)\leq n^{2}-n$. [duplicate]

Let $A$ be an arbitrary $n\times n$ matrix over $\mathbb{C}$ and define the linear transformation $T:F^{n\times n}\rightarrow F^{n\times n}\text{,}\quad T(B)=AB-BA$. I want to prove that $\dim \...
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Can a $2 \times 3$ matrix be full rank?

I have been watching 3blue1brown's Essence of Linear Algebra series on youtube, and I have a question about $2 \times 3$ matrices. For example: \begin{bmatrix}3&1&4\\1&5&9\end{bmatrix} ...
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Find minimal spanning subset of a set for another set efficiently

Correct me if my terminology is wrong since I'm not very good at formal math. I know minimal spanning set is the smallest set of vectors that spans the row space of a matrix, but how can I restrict ...
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Rank = # of non-zero eigenvalues of a diagonalizable matrix [duplicate]

Is the rank of a matrix equal to the # of non-zero eigenvalues of a diagonalizable matrix?
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What exactly is a unique subspace?

I am trying to solve a problem that is asking for what conditions that make $<A>$ unique. It does not explain what <> represents, but I think it means the column space. A is a matrix where ...
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Rank of a simple matrix

While solving an exercise I arrived to this simple matrix and I want to get its rank It is $$\begin{pmatrix}2\cos 2t\\ \cos t \end{pmatrix}$$ It is written in the solution that the rank is 1, they ...
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Proving rank of $(I_n-{1\over n}A_n)$ is $n-1$ where $A_n$ is $n\times n$ with all entries $1$

We were asked to find a symmetric idempotent matrix $H$ with rank $n-1$ such that if $X$ is a column vector with $n$ observations, then ${1\over n}X^THX$ is the variance of observations in $X$. I ...
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basis of space of all $3\times3$ matrices and basis of space of all $3\times3$ matrices with rank $0, 1$ or $2$

I have a set of all $3\times3$ matrices $B$ that have rank $0, 1$ or $2$ - what is the basis of a subspace that this set generates? I came to the conclusion that this subspace consists of lines, ...
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1answer
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When does exist matrices T and H such that HCE=TE? (all matrices are rectangular)

could you please help me with this question; I want to find out the conditions (necessary and /or sufficient) for the existence of two matrices namely H and T such that the equality HCE=TE holds for ...
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Orthogonal Projection onto Range of Matrix

If $A\in\mathbb{R}^{m\times n}$ and $P_{R\left(A\right)}$ is the orthogonal projection onto the range of $A$, i.e. $R\left(A\right)$, then show that $A^{T}P_{R\left(A\right)}=A^{T}$. So far I have ...
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how to efficiently calculate matrix distances

given a number of Actors A (1..n) and a number of Subjects S (1..m) and a number of Positions P (1..o) and each actor can have one position on one subject P (a,s) and each position can be expressed ...
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For a real $n \times n$ matrix $A$ if $x^tA^tx \geq 0 \;\forall \;x,$ then $Au=0 $ iff $A^tu=0.$

Let $A$ be a real $n \times n$ matrix such that $x^tA^tx \geq 0$ for all $x \in \mathbb{R}^n$. Then how can I show that $Au=0 $ if and only if $A^tu=0$ ? Since $A$ and $A^t$ have the same rank ...