Questions tagged [matrix-rank]
For questions regarding the rank of matrices in linear algebra.
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0answers
5 views
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 $p_{c}(n)$ ...
0
votes
0answers
17 views
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 ...
3
votes
3answers
44 views
$\DeclareMathOperator{\rank}{rank}\rank(A) = \rank(B)$, 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 \...
1
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0answers
36 views
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
$$
U=[u(1), ...,u(T)] \quad X=[x(1), ..., x(T)].
$$
...
1
vote
0answers
14 views
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 ...
1
vote
1answer
29 views
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 ...
1
vote
3answers
49 views
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 $A$ ...
1
vote
0answers
27 views
invertibility of the block Vandermonde matrix
Consider a block Vandermonde square matrix
$$X = \begin{bmatrix}
\mathbf{I} & \mathbf{\Omega}(i_0) & \mathbf{\Omega}(2i_0) & \ldots & \mathbf{\Omega}((2p-1)i_0) \\
\mathbf{I} & \...
3
votes
2answers
39 views
if $A \in C^{2015,2015}$ and $rank(A) < 1000$ proof that $\dim(\ker(A+A^T)) > 15$
I want to solve that thesis:
if $A \in C^{2015,2015}$ and $rank(A) < 1000$ proof that $\dim(\ker(A+A^T)) > 15 $
from the fact that $$\dim(im(A)) = \dim(im(A^T))$$ and
$$ \dim(\ker(A))+\dim(im(...
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votes
1answer
14 views
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 $0 \leq k \leq n-1$:
$$Rank(A_{k+1})\leq Rank(A_k)+2$$
...
0
votes
1answer
30 views
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 ...
0
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0answers
22 views
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 ...
3
votes
2answers
42 views
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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2answers
32 views
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(...
1
vote
1answer
19 views
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$
...
0
votes
2answers
26 views
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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0answers
32 views
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 ...
0
votes
1answer
35 views
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 ...
0
votes
3answers
36 views
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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votes
4answers
66 views
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}$ ...
0
votes
1answer
19 views
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)$...
2
votes
0answers
51 views
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 ...
0
votes
1answer
14 views
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? ...
1
vote
1answer
32 views
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)$ ...
0
votes
3answers
77 views
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 ?
1
vote
1answer
28 views
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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votes
0answers
17 views
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}$ ...
0
votes
1answer
106 views
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 &...
0
votes
1answer
59 views
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 $A$.
...
1
vote
2answers
33 views
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?
...
0
votes
1answer
31 views
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 ...
0
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0answers
18 views
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 ...
0
votes
3answers
22 views
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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votes
0answers
14 views
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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0answers
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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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1answer
34 views
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 ...
1
vote
2answers
62 views
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 ...
8
votes
1answer
117 views
$\operatorname{rank}(A^2)+\operatorname{rank}(B^2)\geq2\operatorname{rank}(AB)$ 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. ...
4
votes
0answers
36 views
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 \...
0
votes
1answer
58 views
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}
...
0
votes
0answers
24 views
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 ...
0
votes
0answers
35 views
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?
0
votes
1answer
26 views
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 ...
1
vote
1answer
39 views
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 ...
1
vote
3answers
87 views
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 ...
0
votes
2answers
67 views
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, ...
0
votes
1answer
17 views
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 ...
0
votes
2answers
41 views
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 ...
0
votes
0answers
15 views
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 ...
3
votes
0answers
61 views
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 ...
