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Questions tagged [matrix-rank]

For questions regarding the rank of matrices in linear algebra.

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Proof of the proposition which says that the column rank of $A$ is equal to the row rank of $A$. (Gilbert Strang's new lecture)

I am watching this new lecture by Gilbert Strang. I have the following question. Let $A = \begin{bmatrix}2&1&3\\3&1&4\\5&7&12\\\end{bmatrix}$. Prof. Strang showed that ...
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3answers
47 views

Rank of a matrix with block matrices?

So I'm trying to solve this - Let $A\in M_{5\times 6}(\mathbb{R})$, $A$ has rank $4$. Let $D=\begin{pmatrix} I_5 & A\\ A^T & 0 \end{pmatrix}$. I need to find the rank of $D$. My solution ...
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1answer
17 views

Determining how to make a matrix have less pivots than columns (Example)

I'm studying the following example and I can't figure out why the answer is what it is: "Find the value of $a$ that will give less that 3 pivots". $$A = \begin{bmatrix} 2 & a & 0 \\...
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1answer
58 views

Showing $A-(I-B)A(I-B)'$ is positive semidefinite

I need to show that for two specific matrices $A$ and $B$, where $A$ is symmetric and positive (semi)definite and $B$ is a projection, the difference $$A-(I-B)A(I-B)^\prime$$ is positive semidefinite. ...
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2answers
41 views

Eigenvalues of $Q=I+2P$

I have tried to do it evaluated option (a). I think it is correct.Can not get the other options.Please help me.
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27 views

What is the dimension of the Four Fundamental Subspaces of a $2\times 2$ matrix A with rank $r(A)=1$?

I am taking the 18.06 Linear Algebra course on MIT OpenCourseWare, and I came across problem 30 on section 3.6 of professor Gilbert Strang's book (Introduction to Linear Algebra). The problem states:...
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17 views

Exercise D.2-8 on p.1226 in “Introduction to Algorithms” by CLRS.

I am reading "Introduction to Algorithms" by CLRS. I cannot solve the last part of the following problem. On p.1226 D.2-8: Prove that for any two compatible matrices $A$ and $B$, $$\...
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29 views

About two definitions of the rank of an $m \times n$ matrix $A$ in “Introduction to Algorithms” by CLRS.

I am reading "Introduction to Algorithms" by CLRS. There are the following two definitions of the rank of a matrix $A$ in this book. And the two definitions are equivalent. I proved that, but ...
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1answer
15 views

Existence of left or right pseudoinverse

I'm a bit confused about the following question: Given a matrix $A \in \mathbb{R}^{2 \times 3}$, which one of the left or right pseudoinverse does exist? I know that the left pseudoinverse exists, ...
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23 views

Matrix Equation $W(W^TW)=W(W_1^T W_1)$.

Let $W,W_1\in\mathbb{R}^{m\times d}$ matrices, with $m>d$, and $$ W(W^TW)=W(W_1^T W_1). $$ Suppose that $W_1$ is fixed. Can we characterize all $W'$s obeying the condition? For instance, if ${\rm ...
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1answer
33 views

Rank of the matrices

I want to ask, if rank of matrix with right side (Ab=3), is greater than the rank of the matrix without it (A=2) does it mean that matrix does not have solution? Thanks
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1answer
49 views

Finding matrix with lowest possible rank

Find the values of $x$ for which the matrix \begin{bmatrix} x & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & x \\ \end{bmatrix} has the lowest rank. Since ...
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0answers
19 views

The singular values of the best rank-$k$ approximation to a matrix

Let $A\in\mathbb{C}^{m\times n}$ be a complex matrix. Let $B_k$ be a best rank-$k$ approximation to $A$ such that \begin{equation*} B_k\in\arg\min\limits_{{\rm rank}(B)=k}||A-B||_F, \end{equation*} ...
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23 views

optimality conditions for matrix

I'm struggling with the follwoing exercise: $$ \operatorname*{Minimize}_{x\in\mathbb{R}^n}\frac{1}{2} \|Ax-b\|_2^2 $$ where $A \in \mathbb{R}^{m\times n}, b \in \mathbb{R}^m$ i) Let $ m \geq n$ ...
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29 views

Rank of a matrix with block matrices

So, given $A\in M_{3{\times}4}(R)$ has rank $2$, I built a new matrix - $M=\begin{pmatrix} A& 0& A\\ 0& A& 4A\\ 0& 0& 0 \end{pmatrix}$ Can I claim, that since the rank ...
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29 views

The eigenvalues of a generalized eigenvalue problem

I would like to determine the eigenvalues of a generalized eigenvalue problem (GEP), where $\mathbf{A}=[\array{\mathbf{0}\quad \mathbf{A}_0\\ \mathbf{A}_0 \quad \mathbf{A}_1}]$ and $\mathbf{B}=[\...
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0answers
50 views

Lower bound on k-rank approximation of matrix

I am trying to find a lower bound on $\sum_{l=1}^{m}\| A_l - W\Lambda_lW^T\|_F^2$ where $A \in S_{++}^{n \times n}$ with diagonal elements 1, $W \in R^{n \times k}$ and $\Lambda_l \in R^{k \times k}$ ...
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2answers
82 views

Find the rank of the following matrix

I am thinking on the following problem and have some results but I need a little help. Can some one give me a little hint? Suppose $a_{ij}= \cos (i +j )$ in the matrix $A$ find $\operatorname{rank}(A)...
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11 views

rank of concatenated matrices multiplied by a scalar

Let $\alpha_1,...,\alpha_n \in R^n$. Let the matrices $M_1,...,M_n \in R^{p \times q}$ for some $p,q$. If $\begin{bmatrix} \alpha_1 M_1 \\ \vdots \\ \alpha_n M_n \end{bmatrix}$ is full column rank, ...
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2answers
64 views

Why does every $ m \times n$ matrix of rank $r$ reduce to $(m \times r)$ times $(r \times n)$?

How can I prove the following statement? Every $m \times n$ matrix of rank $r$ reduces to $(m \times r)$ times $(r \times n)$: $A = $ (pivot columns of $A$) (first $r$ rows of $R$) = (COL)(...
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29 views

Finding possible Jordan forms of a matrix

Find the Jordan forms of a matrix $A$ subject to the following conditions: the characteristic polynomial is $(x-1)^4(x+3)^5$. matrix $A-I$ has nullity $4$ and matrix $A+3I$ has nullity $1$. ...
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16 views

What is the lower bound of the rank of a submatrix?

Consider the following statement: If $A$ is an $m\times n$ matrix, and $B$ is a $(m-k)\times n$ submatrix of$A$ formed by deleting $k$ rows of $A$, then $Rank(B)\geq Rank(A)-k$ I came up with ...
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1answer
90 views

How to prove this rank inequality?

Let $n\geq2$ and $A,B\in M_{n}(\mathbb{C})$ such that $B^2=B$. Prove that $$\mbox{ rank }(AB-BA)\leq\mbox{ rank }(AB+BA).$$ If $B$ is zero or the identity matrix, we are done. But $B$ will always be ...
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1answer
25 views

Are the matrices of nullity at most one and non-negative determinant a submanifold with boundary?

Let $\text{GL}_n^+$ be the group of real invertible $n \times n$ matrices with positive determinant. Set $S=\text{GL}_n^+ \cup \{A \, | \, \text{rank} (A) = n-1 \}$. Does $S$ form a a submanifold ...
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0answers
25 views

Deleting entries in binary matrix to reduce rank

A personal research question led me to the following problem. Suppose a randomly-chosen $n\times n$ matrix over $\mathbb{F}_2$ such that: The main diagonal is all 1's. The weight of each row (i.e. ...
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12 views

Rank of tensor like operation?

What is this matrix $M$ called? What is its rank? Take linearly independent vectors $v_1,\dots,v_n\in\mathbb R^m$ and consider the matrix $M$ with $n^n$ rows $v_{i_1}\dots v_{i_n}\in\mathbb R^{mn}$ ...
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5 views

Counting the number of variables and constraints

An optimization problem is defined a graph of V nodes and E edges. First, I defined variables on each nodes (two independent variables for each node), and each edge imposes one constraint (on the 2 ...
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2answers
48 views

Find matrix rank

I have a following matrix: \begin{bmatrix}0&0&-1&5\\0&0&-3&8\\0&0&1&2\\\end{bmatrix} After: Multiply first two rows by -1 and add first and third row together I ...
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1answer
60 views

Why is $(A+BK, B)$ controllable when $(A,B)$ is, but $(C, A+BK)$ is not observable when $(C,A)$ is?

I attempted to use the Popov-Belevitch-Hautus test to prove these, namely $$[A+BK - \lambda I, B]=[A-\lambda I, B]\begin{bmatrix}I & 0\\K & I \end{bmatrix}$$ which are both full rank and ...
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14 views

Definition of the rank of $x_t$ among $(x_1,…,x_T)$

I recently came across the following statement, but I don't understand how to implement it: Let ${x_1,...,x_T}$ be $T$ i.i.d observations of first differences of a variable $x_t$, and let $r(x)$ ...
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187 views

Intuitive explanation of the rank-nullity theorem [duplicate]

I understand that if you have a linear transformation from $U$ to $V$ with, say, $\operatorname{dim} U = 3$, $\operatorname{rank} T = 2$, then the set of points that map onto the $0$ vector will lie ...
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0answers
9 views

Relationship between $\text{rank} ABC $ and $\text{rank} AC$ where $B$ has full rank

Let $A,B,C$ be complex-valued $n \times n$ matrices, where $\text{rank}(B) = n$. I know that for generic $A,B,C$ there is no relationship between $\text{rank}{(ABC)}$ and $\text{rank}(AC)$ like the ...
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1answer
89 views

Finding rank-$1$ matrix

Let $$S = \frac{1}{12} \begin{pmatrix} 1 & 10 & 1 \\ 5 & 2 & 5\\ 1 & 2 & 9\end{pmatrix}$$ Find a rank-$1$ matrix $R$ so that $$ M = S + R $$ will have the same eigenvalues as $...
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1answer
44 views

Question regarding rank of a product of matrices

Let $A$ and $B$ be $5\times5$ matrices. For each $k$, $0\leq k\leq5$, find all possible values for $Rank(BA)$ given that $Rank(AB)=k$. Prove your statement. My attempt: Using the following two ...
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2answers
34 views

Rank of $2I_n-J_n$ where $I$ is the identity matrix and J is a matrix of ones

I am to prove that the following matrix $2I_n - J_n$ has rank $n$. Where $I_n$ is the $n\times n$ identity matrix and $J_n$ is an $n\times n$ matrix of ones. I cannot provide a formal definition for ...
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16 views

Let $P$ be a $n\times n$ matrix, if there is $k\in \mathbb Z^+$ such that $P^k=O$, prove that $P^n=O$. [duplicate]

Let $P$ be a $n\times n$ matrix, if there is $k\in \mathbb Z^+$ such that $P^k=O$, prove that $P^n=O$. I have thought about characteristic polynomial, but it doesn't give me much information. So I ...
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1answer
22 views

Singular points of a matrix when the entries are restriced to a Lie Group

Let $\mathsf{SO}(3)$ be the set of $3 \times 3$ rotation matrices. Let $R\in\mathsf{SO}(3)$ and $r_{ij}$ represent the entry of $R$ sitting at the $i^{th}$ row and $j^{th}$ column, i.e., $$ R \in\ \...
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21 views

In spectrahedra, are minimal rank points always extreme points?

Consider a matrix $A$ in a spectrahedron $S$ such that $$\mbox{rank}(A) \leq \mbox{rank}(B)$$ for all $B\in S$ and assume that at least one matrix $C \in S$ we have that $\mbox{rank}(A) < \mbox{...
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1answer
50 views

Is a full rank square matrix necessarily a positive definite matrix?

(Skagestad, 2005) states the following conclusion in page 128. "The system $(A, B)$ is state controllable if and only if the Gramian matrix $W(t)$ has full rank (and thus is positive definite) for ...
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1answer
23 views

A function that takes vector to matrix of a vector

In Sheldon Axler's Linear Algebra Done Right, when proving that the dimension of range T equals the column rank of $\mathcal{M}(T)$ (where $\mathcal{M}(T)$ is the matrix of the linear map T), it says ...
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1answer
24 views

Relation between rank, nullity and orthogonal complement

Question Let $A\in\mathbb{R}_{d\times d}$ be some square, non-invertible matrix. Prove that if $b\perp\ker(A^T)$, then the non-homogenous system $Ax=b$ has $\infty$ solutions. Background I got ...
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1answer
40 views

Rank of matrix with diagonal 0, and others +-1

Let $B$ be a $(n-1)×(n-1)$ matrix such that: all elements on diagonal equal $0$; and all other either $1$ or $\text{-}1$. Let $A = \begin{bmatrix}B&(1,...,1)^T\\ (1,...,1)&1\end{bmatrix}$, ...
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4answers
60 views

Can a matrix have a rank of $0$?

It's impossible for a matrix to rank $0$, right? Is $\mbox{rank} A > 0$ a known fact?
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1answer
35 views

Generalization of sum of outer product

Consider a matrix $A \in \mathbb{R}^{d \times m}$ such that $m \geq d$ and denote its columns i.e $A_{:, i}$ by $a_i$. Let $AA^T$ is invertible. Now, consider the sum $S(A) = \sum_{r=1}^m a_r a^T_r$...
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2answers
92 views

Proving that determinant is zero

Let $A,B \in M_3(\mathbb{C})$ such that $(AB)^2 = A^2B^2$ and $(BA)^2 = B^2A^2$. Prove that $\det(AB-BA) = 0$.
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1answer
80 views

Similar Matrices have the same rank

Prove that :- If $2$ matrices $A$ and $B$ are similar then they will have the same rank. Proof is given here but I can't understand both answers which are related to image and kernel. I have seen ...
4
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1answer
78 views

How to show that the given matrix has non-zero determinant

Given $p,q$ to be primes where $p<q$ . Show that the following marix has non-zero determinant, \begin{bmatrix} 1&2 & 2 & 2 &\dotso & 2\\ 2&q-p+1 & 1 & 1 &\...
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3answers
98 views

Prove that if a vector space has dimension n then any n + 1 of its vectors are linearly dependent. (Linear Algebra )

I cannot seem to figure out how to prove the following: "Prove that if a vector space has dimension $n$ then any $n + 1$ of its vectors are linearly dependent." I reckon applying proof by ...
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1answer
18 views

Calculating the Kernel, dimension of linear equations, real numbers and galois field

Given these problems below, how would one calculate the result? By intuition, I have managed to solve two of them but cannot crack the last one. Btw, I am not sure that the approach of my ...
0
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1answer
41 views

Given $\textbf{P} = \textbf{X}(\textbf{X}^{\prime}\textbf{X})^{-1}\textbf{X}^{\prime}$, prove that $\mathcal{C}(\textbf{P}) = \mathcal{C}(\textbf{X})$

If $\textbf{X}\in\textbf{R}^{n\times p}$ has full rank ($n\geq p$), so that $\textbf{P} = \textbf{X}(\textbf{X}^{\prime}\textbf{X})^{-1}\textbf{X}^{\prime}$, prove that $\mathcal{C}(\textbf{P}) = \...