Ask Your Question
1

How to construct a class of matrices satisfying a given matrix equation.

asked 2021-04-04 16:06:55 +0200

anonymous user

Anonymous

updated 2021-04-05 07:28:12 +0200

image description

Here we know that $A=I_n$ satisfies the given matrix equation. But can we find other non trivial matrix ($\neq I_n$). In other words, can we construct a class of matrices satisfying the given matrix equation.

Please help regarding this.

edit retag flag offensive close merge delete

Comments

Please help..here p,q,r,n are all given to us..

Kuldeep gravatar imageKuldeep ( 2021-04-04 22:59:58 +0200 )edit

How are graphs related to this matrix equation?

John Palmieri gravatar imageJohn Palmieri ( 2021-04-05 02:34:58 +0200 )edit

If $A$ satisfies $A^{p^2} = A^{q^2} = I_n$, then that works, e.g. if $A^2=I_n$ and $p$ and $q$ are both even. If you want Sage to solve the general problem, then you can write $A$ with all symbolic entries and try to solve the resulting $n^2$ Diophantine equations. If $n$, $p$, or $q$ are large, I think this will be very difficult. What values do you have in mind?

John Palmieri gravatar imageJohn Palmieri ( 2021-04-05 19:06:59 +0200 )edit

but p and q both can not be even since r is given to be odd and they form Pythagorean triple

Kuldeep gravatar imageKuldeep ( 2021-04-05 19:25:57 +0200 )edit

Yes, you're right, but the rest of my comment still applies. For example if $A^3=I$ and both $p$ and $q$ are divisible by 3. Or do you want to assume that $p$, $q$, and $r$ are relatively prime? In any case, if you imagine that $A$ has indeterminate entries and $p=5$, then you will have polynomial equations of degree 25. How would you hope to solve this?

John Palmieri gravatar imageJohn Palmieri ( 2021-04-05 20:28:42 +0200 )edit

1 Answer

Sort by ยป oldest newest most voted
2

answered 2021-04-06 21:33:33 +0200

Max Alekseyev gravatar image

updated 2021-04-07 22:13:31 +0200

The given matrix equation implies that the minimal polynomial of $A$ divides $f(x):=p^2 x^{p^2} - q^2 x^{q^2} - r^2$. It follows that $A$ can be constructed as a block diagonal matrix $\begin{bmatrix} C_g & 0\\ 0 & I_{n-d} \end{bmatrix}$ for any monic divisor $g(x)\mid f(x)$ of degree $d:=\deg g(x)\leq n$, where $C_g$ is the companion matrix of $g(x)$.

Here is a sample code that constructs and prints such matrices:

def compA(n,p,q,r):
    assert p^2 == q^2 + r^2
    R.<x> = PolynomialRing(ZZ)
    f = p^2*x^(p^2) - q^2*x^(q^2) - r^2
    for g in divisors(f):
        if g.degree()>n or not g.is_monic():
            continue
        A = block_diagonal_matrix(companion_matrix(g), identity_matrix(n-g.degree()))
        print(A)

More generally, $A$ can be taken as a block diagonal matrix with blocks $C_{g_1}, \dots, C_{g_k}$, where each $g_i(x)$ is a monic divisor of $f(x)$, and $\sum_{i=1}^k \deg g_i(x) = n$. Matrices similar to such $A$ will also satisfy the given matrix equation.

edit flag offensive delete link more

Comments

Thank you for your answer. But there is no output of this programme when I compile it in Sage

Kuldeep gravatar imageKuldeep ( 2021-04-07 12:18:22 +0200 )edit

The code defines a function. To call it, one needs to provide values of n, p, q, r, like compA(7,5,4,3).

Max Alekseyev gravatar imageMax Alekseyev ( 2021-04-07 13:47:33 +0200 )edit

def compA(7,5,4,3): assert 5^2 == 4^2 + 3^2 R.<x> = PolynomialRing(ZZ) f = 5^2x^(5^2) - 4^2x^(4^2) - 3^2 for g in divisors(f): if g.degree()>7: continue A = block_diagonal_matrix(companion_matrix(g), identity_matrix(7-g.degree())) print(A)

Is it correct...but still the error message is coming

Kuldeep gravatar imageKuldeep ( 2021-04-07 16:25:18 +0200 )edit

I'm not sure what error you're talking about. Click here for an example computing compA(7,15,12,9) at SageCell.

Max Alekseyev gravatar imageMax Alekseyev ( 2021-04-07 22:09:26 +0200 )edit

Sorry, I was making a mistake.. It is now coming very nicely. Thank you

Kuldeep gravatar imageKuldeep ( 2021-04-07 23:02:52 +0200 )edit

Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

Question Tools

1 follower

Stats

Asked: 2021-04-04 16:06:55 +0200

Seen: 101 times

Last updated: Apr 07