# Questions tagged [matrix-rank]

For questions regarding the rank of matrices in linear algebra.

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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 ...
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### 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 ...
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### 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 ...
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### 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 (...
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### 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 ...
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### 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. $\sigma(A) = \{1, -2\}$ ...
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### $\operatorname{rank}(A^2)+\operatorname{rank}(B^2)\geq2\operatorname{rank}(AB)$ 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. ...
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### 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 ...
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### 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 $\mathrm{rank}(AB)=0$.
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### 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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### "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 ...
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### 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 ...
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### 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}(...