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Questions tagged [matrix-rank]

For questions regarding the rank of matrices in linear algebra.

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185 votes
24 answers
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Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix

I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof on Wikipedia and I understand the proof, but I don't "get it". Can someone ...
hari_sree's user avatar
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137 votes
4 answers
107k views

Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$

How can I prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$? This is an exercise in my textbook associated with orthogonal projections and Gram-Schmidt process, but I ...
jaynp's user avatar
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104 votes
4 answers
69k views

Importance of matrix rank

What is the importance of the rank of a matrix? I know that the rank of a matrix is the number of linearly independent rows or columns (whichever is smaller). Why is it a problem if a matrix is rank ...
0x0's user avatar
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93 votes
2 answers
123k views

What is the relation between rank of a matrix, its eigenvalues and eigenvectors

I am quite confused about this. I know that zero eigenvalue means that null space has non zero dimension. And that the rank of matrix is not the whole space. But is the number of distinct eigenvalues (...
Shifu's user avatar
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70 votes
5 answers
113k views

Is the rank of a matrix the same of its transpose? If yes, how can I prove it?

I am auditing a Linear Algebra class, and today we were taught about the rank of a matrix. The definition was given from the row point of view: "The rank of a matrix A is the number of non-zero ...
Vivi's user avatar
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70 votes
9 answers
40k views

How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?

Let $A$ and $B$ be two matrices which can be multiplied. Then $$\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B)).$$ I proved $\operatorname{rank}(AB) \...
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46 votes
6 answers
44k views

Sylvester rank inequality: $\operatorname{rank} A + \operatorname{rank}B \leq \operatorname{rank} AB + n$ [duplicate]

If $A$ and $B$ are two matrices of the same order $n$, then $$ \operatorname{rank} A + \operatorname{rank}B \leq \operatorname{rank} AB + n. $$ I don't know how to start proving this inequality. I ...
Ben Ward's user avatar
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46 votes
3 answers
78k views

A rank-one matrix is the product of two vectors

Let $A$ be an $n\times m$ matrix. Prove that $\operatorname{rank} (A) = 1$ if and only if there exist column vectors $v \in \mathbb{R}^n$ and $w \in \mathbb{R}^m$ such that $A=vw^t$. Progress: I'm ...
coconutbandit's user avatar
44 votes
5 answers
63k views

Proving: "The trace of an idempotent matrix equals the rank of the matrix"

How could we prove that the "The trace of an idempotent matrix equals the rank of the matrix"? This is another property that is used in my module without any proof, could anybody tell me how to prove ...
Quixotic's user avatar
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40 votes
5 answers
132k views

How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$? [duplicate]

How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?
Maysam's user avatar
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37 votes
4 answers
25k views

Why is minimizing the nuclear norm of a matrix a good surrogate for minimizing the rank?

A method called "Robust PCA" solves the matrix decomposition problem $$L^*, S^* = \arg \min_{L, S} \|L\|_* + \|S\|_1 \quad \text{s.t. } L + S = X$$ as a surrogate for the actual problem $$L^*, S^* =...
blubb's user avatar
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36 votes
1 answer
2k views

Rank of a $n! \times n$ matrix

This question is about showing that $n!$ points resulting from applying a function (defined below) to the permutations of $n$ numbers lie on a $n-1$ dimensional hyperplane. Let $X=\langle x_1,\cdots,...
34 votes
1 answer
12k views

Proof that determinant rank equals row/column rank

Let $A$ be a $m \times n$ matrix with entries from some field $F$. Define the determinant rank of $A$ to be the largest possible size of a nonzero minor, i.e. the size of the largest invertible square ...
spin's user avatar
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31 votes
1 answer
7k views

Expected rank of a random binary matrix?

Recently a friend stumbled across this question: Let $M$ be a random $n \times n$ matrix with entries in $\{0,1\}$ (both zero and one has probability $p = q = \frac{1}{2}$). What is its expected ...
dtldarek's user avatar
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26 votes
6 answers
20k views

Rank of sum of rank-$1$ matrices

If you sum a certain number of rank-$1$ matrices: $$X = u_1 u_1^T + u_2 u_2^T + \cdots + u_N u_N^T$$ Is the result guaranteed to be rank-$N$ assuming the individual $U$ vectors are linearly ...
gct's user avatar
  • 583
25 votes
3 answers
90k views

Relation between determinant and matrix rank

Let $A$ a square matrix with the size of $n \times n$. I know that if the rank of the matrix is $<n$, then there must be a "zeroes-line", therefore $\det(A)=0$. What about $\text{rank}(A)=n$? ...
AndrePoole's user avatar
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25 votes
2 answers
13k views

Show $\operatorname{rank}(A) + \operatorname{rank}(B) \ge \operatorname{rank}(A+B)$ [duplicate]

Show $\operatorname{rank}(A) + \operatorname{rank}(B) \ge \operatorname{rank}(A+B)$, where $A,B \in M_{m\times n}(\mathbb{F})$. I'm trying to think in terms of linear transformations. We can define $...
AnnieOK's user avatar
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23 votes
5 answers
20k views

Show that $\operatorname{rank}(A+B) \leq \operatorname{rank}(A) + \operatorname{rank}(B)$

I know about the fact that $\operatorname{rank}(A+B) \leq \operatorname{rank}(A) + \operatorname{rank}(B)$, where $A$ and $B$ are $m \times n$ matrices. But somehow, I don't find this as intuitive as ...
xenon's user avatar
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22 votes
3 answers
70k views

Rank product of matrix compared to individual matrices. [duplicate]

Possible Duplicate: How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$? If $A$ is an $m\times n$ matrix and $B$ is a $n \times r$ matrix, prove that the rank of matrix $...
mello's user avatar
  • 236
21 votes
4 answers
3k views

Show that B is a nonsingular matrix (not that obvious).

How would you proceed if you were asked in an interview to show that B is a nonsingular matrix (in an elegant way)? $$B= \begin{pmatrix} 1& 1.25& −0.50& 0.15\\ 0.15& 2& 1.25& −...
QFi's user avatar
  • 1,200
20 votes
3 answers
24k views

Proof that the rank of a skew-symmetric matrix is at least $2$

Is there a succinct proof for the fact that the rank of a non-zero skew-symmetric matrix ($A = -A^T$) is at least 2? I can think of a proof by contradiction: Assume rank is 1. Then you express all ...
Naga's user avatar
  • 631
20 votes
7 answers
21k views

Relation between trace and rank for projection matrices

If $A $ is an $n \times n$ matrix over $\mathbb C$ such that $A^2=A$ then is it true that $\operatorname{trace} A = \operatorname{rank} A$?
Learnmore's user avatar
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19 votes
4 answers
19k views

Is the rank of a matrix equal to the number of non-zero eigenvalues?

I have studied before that the rank of a matrix = number of non zero Eigen values. But recently i came across a problem and i dont think it is valid there. I know i am going wrong somewhere. $$A= \...
Shubham's user avatar
  • 473
19 votes
2 answers
16k views

Rank of the outer product of two vectors

I have come across the statement that the rank of the outer product of two vectors is always $1$, but why is that true?
user avatar
19 votes
2 answers
3k views

Prove that $\det(AB-BA)=0$

Let $A,B$ be two $3 \times 3$ matrices with complex entries such that $$(A-B)^2=O_3$$ Prove that $$\det(AB-BA)=0$$ I tried to prove this with ranks. I denoted $X=A-B$ and thus $X^2=O_3$ which means ...
AndrewC's user avatar
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18 votes
2 answers
466 views

Uniqueness of trace as linearization of the rank

It is not difficult to show that if $A \in M_n(k)$ for some field $k$, and $A^2=A$ then $$\operatorname{tr}(A) = \dim(\operatorname{Im}(A))$$ In this comment, Terry Tao wrote: This property, together ...
Alphonse's user avatar
  • 6,352
18 votes
2 answers
561 views

$\operatorname{rank} A = \operatorname{rank} A^2$ if and only if $\lim_{\lambda \to 0} (A+\lambda I)^{-1}A$ exists

Let $A \in \mathbb C^{n \times n}$. Prove that $\operatorname{rank} A = \operatorname{rank} A^2$ if and only if $\displaystyle\lim_{\lambda \to 0} (A+\lambda I)^{-1}A$ exists. I am stuck on this ...
meiji163's user avatar
  • 3,959
17 votes
3 answers
20k views

Why do similar matrices have the same rank?

I have seen some proofs on the Internet, which make use of the transformation map. However, I couldn't understand the methods since what I learned about the transformation map is so superficial. Can ...
JimfyWinsy's user avatar
17 votes
2 answers
11k views

Diagonalizable vs full rank vs nonsingular (square matrix)

There are many discussions of such type problems (comparison), for example: Diagonalizable vs Normal Today, I want to clearly understand the topic. Suppose the matrix $A\in \mathbb{R}^{n\times n}$. ...
sleeve chen's user avatar
  • 8,303
15 votes
5 answers
13k views

Why do elementary matrix operations not affect the row space of a given matrix?

I have shown that two of the three elementary operations will not change the image of the row space of the matrix: given a row vector $\vec{v}$, $k\vec{v}$ will span the same (scalar multiplication), ...
LinAlgStudent's user avatar
15 votes
1 answer
16k views

Connection between rank and positive definiteness

I would like to know, is there a connection between the rank of a matrix and whether it is positive definite? Specifically, if I can prove that a matrix is not full rank, then can I say that it is not ...
space_voyager's user avatar
14 votes
1 answer
275 views

Characterization of the subspaces of $\mathbb R^{m\times n}$ induced by rank-1 matrices?

Consider a linear subspace $S$ of the space of $m\times n$ real valued matrices $\mathbb R^{m\times n}$. When does $S$ admit a basis consisting only of rank-1 matrices? I.e. is there a simple ...
Hyperplane's user avatar
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13 votes
5 answers
15k views

Prove Sylvester rank inequality: $\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n$

If $A$ is a $m \times n$ matrix and $B$ a $n \times k$ matrix, prove that $$\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$$ Also show when equality occurs.
Christmas Bunny's user avatar
13 votes
1 answer
2k views

What is the rank of the matrix consisting of all permutations of one vector? [duplicate]

Let $a=(a_1,...,a_n)^\top\in\mathbb{R}^n$ be a column vector and let $M_1,...,M_{n!}$ denote all $n\times n$ permutation matrices. When is the rank of the matrix that consists of all possible ...
MathKi's user avatar
  • 183
13 votes
2 answers
10k views

The "range" and "image" of a transformation refer to the same thing, right?

I'm hoping this is true because I've been told that "the rank of a transformation is the dimension of its image", and also that "the rank of a transformation is the dimension of its range". This doesn'...
James Ronald's user avatar
  • 2,331
13 votes
2 answers
12k views

Is the number of linearly independent rows equal to the number of linearly independent columns?

For any matrix the column rank and row rank are equal. As I understand it rank means the number of linearly independent vectors, where vectors is either the rows or columns of the matrix. This seems ...
Jonathan.'s user avatar
  • 1,525
13 votes
1 answer
2k views

Rank, nuclear and Frobenius norms of a matrix

The nuclear norm, denoted $\| \cdot \|_*$ is a good surrogate for the rank when minimizing problems like $$\label{pb1}\tag{1} \min_X \operatorname{rank} (X) : AX = B $$ Here, we're trying to find a ...
davcha's user avatar
  • 1,745
12 votes
4 answers
100k views

How to calculate the rank of a matrix?

I need to calculate the rank of the matrix $A$, shown below: $$ A= \begin{bmatrix} 3 & 2 & -1\\ 2 & -3 & -5\\ -1 & -4 &- 3 \end{bmatrix} $$ I know that I need to calculate $...
Over Killer's user avatar
12 votes
3 answers
28k views

Rank of matrix $AB$ when $A$ and $B$ have full rank

Let $A$ be a $m \times n$ matrix with rank $n$, and let $B$ be a $n \times p$ matrix with rank $p$. Calculate the rank of matrix $C=AB$. The rank of a matrix is the number of linearly independent ...
Jonny's user avatar
  • 523
12 votes
4 answers
10k views

Solving a system of non-linear equations

Let $$(\star)\begin{cases} \begin{vmatrix} x&y\\ z&x\\ \end{vmatrix}=1, \\ \begin{vmatrix} y&z\\ x&y\\ \end{vmatrix}=2, \\ \begin{vmatrix} z&x\\ y&z\\ \end{...
user avatar
12 votes
3 answers
6k views

Proof of: $AB=0 \Rightarrow Rank(A)+Rank(B) \leq n$

As the title says, am searching for a proof of If $A,B \in \mathbb{R}^{n\times n}$ and $AB=0$ then $\mathrm{rank}(A)+\mathrm{rank}(B) \leq n$ I am doing this as preparation for an upcoming ...
ftiaronsem's user avatar
  • 1,089
12 votes
1 answer
1k views

Multiplicity of 0 eigenvalue of directed graph Laplacian matrix

I am looking for a result (if it exists) for directed graphs relating the multiplicity of 0 eigenvalues of the directed Laplacian matrix. Consider a directed graph $\mathcal{G}=(\mathcal{V},\mathcal{...
yen's user avatar
  • 121
11 votes
7 answers
4k views

When solving for eigenvector, when do you have to check every equation?

For example, suppose we want to solve for the eigenvectors of: $$A = \begin{bmatrix} 0 & 1 \\ 2 & -1 \end{bmatrix} $$ We quickly find the eigenvalues are $1, -2$ i.e. $\sigma(A) = \{1, -2\}$ ...
Olórin's user avatar
  • 5,455
11 votes
1 answer
397 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. ...
Yuxiao Xie's user avatar
  • 8,596
10 votes
2 answers
6k views

Why do row operations not change the column rank?

From this question link, I got to know that row operation (row subtraction and row permutation) do change column space. But still it seems that it does not change the column rank. I am trying to prove ...
DongukJu's user avatar
  • 366
10 votes
2 answers
12k views

If $AB=0$ prove that $\mathrm{rank}(A)+\mathrm{rank}(B)\leq n$

Let $A,B\in M_n(\mathbb{R})$ such that $AB=0$. Prove that $$\mathrm{rank}(A)+\mathrm{rank}(B)\leq n.$$ From the given information, I only know that $\mathrm{rank}(AB)=0$.
Wang Kah Lun's user avatar
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10 votes
2 answers
3k 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 ...
Joseph's user avatar
  • 187
10 votes
1 answer
2k views

"Rank-K Correction" of a matrix and significance?

Today my studies led me to read about the matrix inversion lemma, which Wikipedia introduces as follows: In mathematics (specifically linear algebra), the Woodbury matrix identity, named after Max ...
Benjamin Bray's user avatar
10 votes
1 answer
242 views

Dimension of a Subspace of $\text{Hom}_\mathbb{K}(\mathcal{V},\mathcal{W})$ Consisting of Only Linear Transformations of Rank $\leq r$

Background. I have a conjecture (stated in two ways below), which I would like to see whether it is true (both the inequality part and the part involving the equality cases). The motivation comes ...
Batominovski's user avatar
  • 49.7k
10 votes
0 answers
197 views

Does the space of matrices above rank $k$ admit a transitive Lie group action?

$\newcommand{\End}{\operatorname{End}}$ $\newcommand{\GL}{\operatorname{GL}}$ Let $V$ be a real $d$-dimensional vector space ($d \ge 4$). Let $2 \le k \le d-2$. Define $H_{>k}=\{ A \in \text{End}(...
Asaf Shachar's user avatar
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