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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 $f$.

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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 help me out with this ? I find it hard to wrap my head around…
hari_sree
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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 am unsure how they are relevant.
jaynp
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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 deficient? Also, why is the smaller value between…
0x0
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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 ( thus independent eigenvectos ) is the rank of…
Shifu
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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 rows in the reduced row-echelon form of A". The…
Vivi
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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 $\operatorname{rank}(AB) \leq \operatorname{rank}(B)$ by interpreting $AB$…
user365
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Sylvester rank inequality: $\operatorname{rank} A + \operatorname{rank}B \leq \operatorname{rank} AB + n$

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 would be very pleased if someone helps me.…
Ben Ward
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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 this one?
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How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?

How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?
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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 $X=\langle x_1,\cdots,x_n \rangle$ and $Y=\langle y_1,\cdots,y_n…
Helium
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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 going back and forth between using the definitions…
coconutbandit
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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^* = \arg \min_{L, S} rank(L) + \|S\|_0 \quad…
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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 rank? My intuition is that it would be something of…
dtldarek
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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 submatrix of $A$. It is true that the…
spin
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Show $\operatorname{rank}(A) + \operatorname{rank}(B) \ge \operatorname{rank}(A+B)$

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 $T_a, T_b:\mathbb{F}^n\rightarrow…
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