Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [matrix-rank]

For questions regarding the rank of matrices in linear algebra.

In linear algebra, the rank of a matrix $A$ is the size of the largest collection of linearly independent columns of $A$ (the column rank) or the size of the largest collection of linearly independent rows of $A$ (the row rank). For every matrix, the column rank is equal to the row rank. It is a measure of the “nondegenerateness” of the system of linear equations and linear transformation encoded by $A$. There are multiple definitions of rank. The rank is one of the fundamental pieces of data associated with a matrix.

If $A$ is the matrix of a linear map $f$, then the rank of $A$ is equal to the dimension of the image of $f$.

2509 questions
1
vote
0 answers

Rank of a matrix sum: equality case

Let $A\in\mathbb{R}^{m\times m} $ and $B\in\mathbb{R}^{m\times m}$. It is known that $rank(A+B)\leq rank(A)+rank(B)$. My question is when does equality holds for this inequality? What is the condition for the case of more than 2 matrices?
DnzSnl
  • 21
1
vote
3 answers

Prove matrix $uu^T+vv^T$ has rank two.

Can somebody help me in proving that for a given two non-zero column vectors $u$ and $v$ in $\mathbb{R}^n$, such that $u$ and $v$ are linearly independent, then the matrix $uu^T+vv^T$ has rank two.
Uday Kumar
  • 111
0
votes
1 answer

Rank of a block matrix

Given rank(A)=k, how do we prove that rank(B)=2k if $ B= \left[ {\begin{array}{cc} A & 0 \\ 0 & A \\ \end{array} } \right] $?
aaka
  • 1
  • 2
  • 10
0
votes
1 answer

A is a $2 \times 2$ matrix. Show that there exist $2 \times 2$ matrices with certain rank requirements.

A is a $2 \times 2$ matrix. Show that there exist $2 \times 2$ matrices $B_1$ and $B_2$ such that $\textrm{rank}(A+B_1)=1$ and $\textrm{rank}(A+B_2)=2$. I can write examples of $B_1$ and $B_2$, but could anyone give me a more thorough answer?…
Henry Ma
  • 33
0
votes
1 answer

Help for proving rank of matrix

This is a problem from our practice exam. Could anyone tell me how to approach this question and prove details. Super appreciate!
Guess
  • 57
0
votes
0 answers

Rank of a matrix-valued function defined on a neighborhood of a point

I am reading "Ordinary Differential Equations" by L. S. Pontryagin (translated from Russian to Japanese). Suppose $u^1(x^1,\dots,x^n),u^2(x^1,\dots,x^n),\dots,u^k(x^1,\dots,x^n)$ are functions of class $C^1$ defined on a neighborhood of a point…
佐武五郎
  • 886
0
votes
1 answer

Rank of a matrix in trapezoidal form

Is it true that, to determine the rank of a matrix in trapezoidal form (row echelon form but pivots can be 0), you need to count the number of rows (or columns) with the same number of zeroes, except the null rows (columns)? For example, take the…
Neox
  • 717
0
votes
0 answers

rank(A) = rank($A^t *A$) for a real matrix A: when is this NOT true?

I found the proof for rank($A$) = rank($A^T \cdot A$) , A $\in \mathbb{R}^{n \times p}$ once here, Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$ , and also in the wikipedia entry. I work with a $3200x10$ matrix…
Simon
  • 185
  • 5
0
votes
1 answer

If $A$ is a $n\times n$ matrix with entries from $\mathbb R$ or $\mathbb C$ (complex) and ${A^{T}}A=0$ prove $\operatorname{Rank}(A)\le \frac{n}{2}$

If $A$ is a $n\times n$ matrix with entries from $\mathbb R$ or $\mathbb C$ (complex) and ${A^{T}}A=0$ prove that $$\operatorname{Rank}(A)\le \frac{n}{2}.$$ i know how to prove it when the entries are Real numbers but i have trouble when the…
Charis Gios
  • 5
0
votes
1 answer

For any square matrix F, is it possible that $\operatorname{rank}(F F^T-F^T F) = 1?$

Given $F$ is any square matrix, let $F^T$ be the conjugate transpose. Is it possible that $\operatorname{rank}(F F^T-F^T F) = 1?$? It seems that it is impossible, but I don't know how to prove it. On the other hand, is there specific name for the…
Ivan Lg
  • 43
0
votes
0 answers

How to prove that if $A$ and $B$ are $n\times n$ matrices then $\operatorname{rank}(A) + \operatorname{rank}(B) \le \operatorname{rank}(AB) + n$

Prove that if $A$ and $B$ are $n\times n$ matrices then $\operatorname{rank}(A) + \operatorname{rank}(B) \le \operatorname{rank}(AB) + n$
jiabo Li
  • 9
0
votes
1 answer

Does a matrix of rank $r$ always contain an invertible submatrix of size $r\times r$?

If a matrix $A$ (it doesn't need to be a square matrix) has rank $r$, is it always possible to choose $r$ rows and $r$ columns in $A$ such that the matrix $B$ formed by removing the rows and columns that are not selected also has rank $r$, i.e., $B$…
HelloGoodbye
  • 571
  • 2
  • 12
0
votes
1 answer

How to judge the rank relationship of matrix equation $AB=E$

Note: The following questions are from the 5th question of the 2010 Chinese Graduate Mathematical Entrance Examination (first set): Suppose A is an $m*n$ matrix, B is an $n*m$ matrix, and $AB=E$ (E is the m-order unit matrix), then the following…
Please correct GrammarMistakes
  • 399
  • 3
  • 16
0
votes
2 answers

How to prove that the rank of a nonderogatory matrix is at least its order $-1$

If $A \in M_{n}$ is nonderogatory, why is rank $A \geq n-1 ?$
Ben
  • 147
0
votes
0 answers

Relation between rank of matrix and eigenvalues.

Suppose matrix $A$ has rank $k$. If all the eigenvalues of $A^TA$ are real, then is it true that $A^TA$ has $k$ positive eigenvalues(including repeated ones)? Can this be addressed using rank nullity theorem? Appreciate if one could advise me on…
Alexy Vincenzo
  • 5,400
1
2
3 4