## Research level applications of “row rank = column rank”?

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No less an authority than Gilbert Strang frames "row rank equals column rank" (and a couple of other facts) as "The Fundamental Theorem of Linear Algebra."

I'd simply like to assemble (for teaching purposes) a list of research level applications of this basic fact.

Applications can be theoretical or practical, and I would particularly appreciate learning what value this fact has in the minds of physicists.

(It goes without say) please do not start a debate concerning the centrality of this fact (that's not what MO is for). My question merely seek insights or pointers to the literature that would support making a positive case for centrality. So if you use linear algebra all the time but never this fact, no need to chime in.

"Row rank equals column rank" has the consequence for square matrices that $A$ singular makes $A^T$ singular; I'm sure this case comes up everywhere. Here I'm specifically looking for applications of the one-the-nose numerical equality of ranks.

Feel free to offer a philosophical take on linear algebra that would support the centrality of "row rank equals column rank" even if that philosophy isn't grounded in specific results. (For example, does this simple statement offer a hidden paradigm for whole sophisticated theories.)

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Perhaps it's worth remarking that the measuring the failure of this theorem for linear operators on infinite dimensional spaces has lots of applications: this is the basis of Fredholm index theory. – Paul Siegel Feb 28 at 15:29

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(I took the liberty of editing the title because, adding the " " ) – Qfwfq Feb 28 at 17:29

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To me, the statement that column rank is equal to row rank is equivalent to the statement that the dimension of the quotient space of the domain by the kernel is equal to the dimension of the image. Or that the dimension of the domain minus the dimension of the kernel is equal to the dimension of the range minus the dimension of the cokernel. This is used almost everywhere in mathematics, notably homology and cohomology theories. – Deane Yang Mar 1 at 5:44

## 3 Answers

There is this proof of the De Bruijn-Erdös theorem: $p$ points in the plane, not all on the same line, at least $p$ lines go through at least two of the points.

The linear algebraic proof goes like this: let $A$ be the incidence matrix of points versus lines (each row is labeled by a point, each column by a line going through at least two of the points, and the $ij$ coefficient is $1$ if the given point is on the given line, $0$ otherwise). Then it is easily seen that $det(AA^T)\neq0$. In particular the rank of $A$ is $p$, and since this is its column rank the number of columns must be at least $p$.

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beautiful proof!  – Delio Mugnolo Feb 28 at 22:27

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In some sense you can view the singular value decomposition as a sharpening of this theorem (for real and complex matrices, anyway). This, in turn, is useful all over the place.

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Let $M$ be a finite monoid (e.g. a finite group, etc.) and $k$ a field. A function $f$ in the algebra $k[M]$ of $M$ is said of rank $m$ if the dimension of its orbits by shifts (i.e. for $y\in M$; $y^{-1}f,fy^{-1}$ are the "shifted" function defined by $y^{-1}f(x):=f(yx)$, $fy^{-1}(x)=f(xy)$) is of rank $m$. The fact the the "right rank" equals the "left rank" is an incarnation of the equality of the (row-column) ranks by means of the Hankel matrix indexed by $M\times M$ and defined by $$(x,y)\to f(xy)$$
This holds even for infinite monoids when one considers the functions that have finite dimensional orbits by shifts [1] (for example, with $M=(\mathbb{R},+,0)$, the functions $\sin$ and $\cos$ are of rank 2, $\exp$ has rank 1). The interest of this notion is when the monoid is NOT commutative and then, the Hankel matrix may not be symmetric.

[1] For people who are familiar with these matters, this is the Sweedler's dual of $k[M]$ for the comultiplication of the monoid.

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