Testwiki:Reference desk/Archives/Mathematics/2020 August 23

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August 23

bound to "sparsity" of a matrix

Let say I got a set of matrices, with at most k_i non zero values. What is the the upper bound of sparsity of the matrix product of the set? Meaning, how non sparse can it get? What is the upper bound of powering a sparse matrix? --Exx8 (talk) 19:27, 23 August 2020 (UTC)

It can go up pretty quickly. Let the square $m\times m$ matrix $M$ be defined by $M_{i,j}=1$ if $i=1$ or $j=1$, while all other entries are zero. So there are only $2m-1$ nonzero entries. Yet $M^2$ is completely filled with $m^2$ nonzero entries.  --Lambiam 20:32, 23 August 2020 (UTC)

Rectangle with given Y coordinates

(please check the following logic) If I am given the Y coordinates for 4 points (in ascending order a,b,c,d), a rectangle can be made, only if a+d = b+c. Such a rectangle could be (b,a), (a,b), (c,d) , (d,c) which will have one side of length sqrt(2) (c-a) and one of length sqrt(2)(b-a) for a total area of 2*(c-a)*(b-a). Correct?Naraht (talk) 21:10, 23 August 2020 (UTC)

I have not tried all combinations, but for the quadrilateral whose vertices are given by $P=(a,b)$, $Q=(b,a)$, $R=(d,c)$ and $S=(c,d)$, this is true. For this to be a rectangle, the sides $PQ$ and $QR$ have to form a right angle; the rest then follows by symmetry.

We can represent the directions of these sides as vectors: $\overrightarrow{PQ}=(b-a,a-b)$ and $\overrightarrow{QR}=(d-b,c-a)$. The angle is a right angle when these two vectors are orthogonal, which (assuming neither is a zero vector) means that their dot product vanishes:

$(b-a,a-b)\cdot (d-b,c-a)=(b-a)(d-b)+(a-b)(c-a)=0.$

After simplification this becomes $(b-a)((a+d)-(b+c))=0$. So, assuming $a\ne b$, it follows that $a+d=b+c$.  --Lambiam 22:43, 23 August 2020 (UTC)

Given a, b, c, d in any order, with a+d=b+c, not only is it possible to build a rectangle with the given y-coordinates, it's possible to build a square. Namely, (a,c), (b,a) (c,d) (d, b). --RDBury (talk) 02:34, 24 August 2020 (UTC)

What is the area of that square?Naraht (talk) 14:06, 24 August 2020 (UTC)

The length of the diagonal is $D=\sqrt{(d-a{)}^2+(b-c{)}^2}$, so the area is $\frac{D^2}{2}=\frac{(d-a{)}^2+(c-b{)}^2}{2}$. Iffy★Chat -- 14:54, 24 August 2020 (UTC)