Questions tagged [matrix-rank]
For questions regarding the rank of matrices in linear algebra.
2
votes
1answer
52 views
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 ...
0
votes
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 ...
0
votes
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".
$$A =
\begin{bmatrix}
2 & a & 0 \\...
1
vote
1answer
58 views
Showing $A-(I-B)A(I-B)'$ is positive semidefinite
I need to show that for two specific matrices $A$ and $B$, where $A$ is symmetric and positive (semi)definite and $B$ is a projection, the difference
$$A-(I-B)A(I-B)^\prime$$
is positive semidefinite.
...
1
vote
2answers
41 views
Eigenvalues of $Q=I+2P$
I have tried to do it evaluated option (a). I think it is correct.Can not get the other options.Please help me.
0
votes
0answers
27 views
What is the dimension of the Four Fundamental Subspaces of a $2\times 2$ matrix A with rank $r(A)=1$?
I am taking the 18.06 Linear Algebra course on MIT OpenCourseWare, and I came across problem 30 on section 3.6 of professor Gilbert Strang's book (Introduction to Linear Algebra).
The problem states:...
0
votes
0answers
17 views
Exercise D.2-8 on p.1226 in “Introduction to Algorithms” by CLRS.
I am reading "Introduction to Algorithms" by CLRS.
I cannot solve the last part of the following problem.
On p.1226
D.2-8:
Prove that for any two compatible matrices $A$ and $B$,
$$\...
0
votes
0answers
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 ...
0
votes
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, ...
1
vote
0answers
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 ...
1
vote
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
4
votes
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 ...
0
votes
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*}
...
0
votes
0answers
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$ ...
0
votes
0answers
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 ...
0
votes
0answers
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}=[\...
2
votes
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 $\Lambda_l \in R^{k \times k}$ ...
1
vote
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)...
0
votes
0answers
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, ...
2
votes
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)(...
1
vote
0answers
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 $1$. ...
0
votes
0answers
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 ...
3
votes
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 ...
1
vote
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 ...
1
vote
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. ...
0
votes
0answers
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 $v_{i_1}\dots v_{i_n}\in\mathbb R^{mn}$ ...
0
votes
0answers
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
0answers
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 $r(x)$ ...
7
votes
2answers
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 ...
0
votes
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 ...
2
votes
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 $...
0
votes
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 ...
1
vote
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 ...
0
votes
0answers
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 ...
1
vote
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.,
$$
R \in\ \...
0
votes
0answers
21 views
In spectrahedra, are minimal rank points always extreme points?
Consider a matrix $A$ in a spectrahedron $S$ such that
$$\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{...
0
votes
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 ...
0
votes
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
...
1
vote
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 ...
2
votes
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 $A = \begin{bmatrix}B&(1,...,1)^T\\
(1,...,1)&1\end{bmatrix}$, ...
1
vote
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?
0
votes
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 $S(A) = \sum_{r=1}^m a_r a^T_r$...
1
vote
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 $\det(AB-BA) = 0$.
0
votes
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
votes
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 &\...
2
votes
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 ...
1
vote
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
votes
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}) = \...
