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janhardo

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12 years, 25 days
B. Ed math
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How do I evaluate a simplified expressio...

janhardo 935
May 05 2026
0 0

How do I evaluate a simplified expression and determine the dimension matrix from module code?

kroneckerproducts_module_with_dimensioncheckDEF_5-5-2026_.mw

Examine the mixed Kronecker expression ?...

janhardo 935
May 04 2026
0 0

Examine the mixed Kronecker expression ?
kroneckerproducts_module_with_dimensioncheck__4-5-2026.mw

Module has dimensioncheck for numeric use and symbolic use a dimension warning

Check this in Linear Algebra Your e...

janhardo 935
May 03 2026
0 0

Check this in Linear Algebra 
Your example is correct , because  X and Y do have the right dimensions

Experiment with dimensions  

restart;
with(LinearAlgebra):

# =========================================================
# Kies afmetingen zodat de regel geldt:
#   X is een kolomvector (n×1)
#   Y is een rijvector   (1×m)
# =========================================================
# A : p × n          (p willekeurig)
# X : n × 1          (kolomvector)
# Y : 1 × m          (rijvector)
# B : m × q          (q willekeurig)
p := 3;
n := 2;
m := 4;
q := 2;

A := Matrix(p, n, symbol = a);
X := Matrix(n, 1, symbol = x);
Y := Matrix(1, m, symbol = y);
B := Matrix(m, q, symbol = b);

# Linker- en rechterlid van de identiteit
L := A . (KroneckerProduct(X, Y)) . B;
R := KroneckerProduct(A . X, Y . B);

# Verschil (moet een nulmatrix zijn)
Verschil := simplify(L - R);
print(Verschil);

@Harry Garst This mixed Kronecker e...

janhardo 935
May 03 2026
0 0

@Harry Garst 

This mixed Kronecker expression is a special case; the same seems to apply to the module code
The module code does not yet check the dimensions of the simplified expressions, but it could.(i  add this later)

The question you’re asking is a tricky one, but hopefully this answer will be helpful
You can use the procedure to check the input for dimensions, and with this knowledge, you can also check this in a linear algebra package.

gemixed_produkt_expressie_voorwaarden_dimensie_controle_en_uitleg_3-5-2026.mw

@Harry Garst You try to check rule ...

janhardo 935
May 03 2026
0 0

@Harry Garst 

You try to check rule 7 ...
I used the Kron notation, easier to read

Download vraag-_Is_this_correct_mprimes_3-5-2026.mw

  > ....

janhardo 935
May 03 2026
0 0


 

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> 

restart;
with(plots): with(plottools):

# ------------------------------------------------------------
# 1. PRINT THE MATHEMATICAL DERIVATION
# ------------------------------------------------------------
print("Why does stacking these triangles produce a logarithmic spiral?");
print("Definition: Z(0) = 8, Z(1) = 8*(1/4 + I*sqrt(3)/4) = 2+2*I*sqrt(3)");
print("Recurrence: Z(n) = Z(n-1) * (1/4 + I*sqrt(3)/4)");
print("Modulus of the factor = 1/2, argument = Pi/3 (60°)");
print("So |Z(n)| = 8*(1/2)^n, arg(Z(n)) = n*Pi/3");
print("In polar coordinates: r = 8*(1/2)^n, theta = n*Pi/3");
print("Eliminate n: n = 3*theta/Pi -> r = 8*exp(-(3*ln(2)/Pi)*theta)");
print("This is a logarithmic spiral. The vertices Z(n) lie exactly on this spiral.\n");

# ------------------------------------------------------------
# 2. DEFINE THE COMPLEX VERTICES Z(n)
# ------------------------------------------------------------
Z := n -> evalf(8 * (1/4 + I*sqrt(3)/4)^n):   # numeric values for plotting

# ------------------------------------------------------------
# 3. PLOT THE LOGARITHMIC SPIRAL (red, thick) through the vertices
# ------------------------------------------------------------
r0 := 8: b := -3*ln(2)/Pi:
spiral := polarplot(r0*exp(b*theta), theta=0..8*Pi,
                    color="Red", thickness=3, scaling=constrained,
                    axes=boxed, labels=["Re(z)", "Im(z)"]):
# ------------------------------------------------------------
# 4. PREPARE THE TRIANGLES (original definition, animated)
# ------------------------------------------------------------
Triangle := i -> polygon([[0,0], [Re(Z(i-1)), Im(Z(i-1))], [Re(Z(i)), Im(Z(i))]]):

frame := proc(n::posint)
    local i, colors;
    colors := ["Red","Green","Blue","Yellow","Magenta","Cyan"];
    display(seq(Triangle(i), i=1..n),
            color=[seq(colors[(i-1 mod 6)+1], i=1..n)],
            transparency=0.5);   # semi-transparent to see the spiral
end proc:

# Create the animation (first 6 frames)
animation := display(seq(frame(i), i=1..6), insequence, scaling=constrained):

# ------------------------------------------------------------
# 5. PLOT THE VERTICES AS BLACK DOTS (to show they lie on the spiral)
# ------------------------------------------------------------
vertices := [seq([Re(Z(n)), Im(Z(n))], n=0..6)]:
vertex_plot := pointplot(vertices, symbol=solidcircle, symbolsize=7, color="Green"):

# ------------------------------------------------------------
# 6. COMBINE EVERYTHING IN A LARGE DISPLAY
# ------------------------------------------------------------
final_plot := display(animation, spiral, vertex_plot,
                      scaling=constrained, size=[1800,900],
                      title="Logarithmic spiral through the vertices of stacked triangles",
                      titlefont=[Times,16]):

# Show the final plot
final_plot;

"Why does stacking these triangles produce a logarithmic spiral?"

  

"Definition: Z(0) = 8, Z(1) = 8*(1/4 + I*sqrt(3)/4) = 2+2*I*sqrt(3)"

  

"Recurrence: Z(n) = Z(n-1) * (1/4 + I*sqrt(3)/4)"

  

"Modulus of the factor = 1/2, argument = Pi/3 (60°)"

  

"So |Z(n)| = 8*(1/2)^n, arg(Z(n)) = n*Pi/3"

  

"In polar coordinates: r = 8*(1/2)^n, theta = n*Pi/3"

  

"Eliminate n: n = 3*theta/Pi -> r = 8*exp(-(3*ln(2)/Pi)*theta)"

  

"This is a logarithmic spiral. The vertices Z(n) lie exactly on this spiral. "

  

 
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> 

 


 

Download logaritmic_spiral_in_complex_planeDEF_3-5-2026.mw

@C_R Interesting, thanks!...

janhardo 935
May 02 2026
0 0

@C_R 

Interesting, thanks for  geodesics_AI.mw

Maple Worksheet - Error Failed to load t...

janhardo 935
April 30 2026
0 0

Download kroneckerproducts_module_next_level__30-4-2026.mw

@C_R   ......

janhardo 935
April 29 2026
0 0

@C_R 
I did a fix 

Download bol_met_grootcirkels_mprimesDEF_29-4-2026.mw

geodetics_curve_for_orb__mprimes_29-4-20...

janhardo 935
April 29 2026
0 0

geodetics_curve_for_orb__mprimes_29-4-2026.mw

@Harry Garst Ja, en hopelijk kan je...

janhardo 935
April 27 2026
0 0

@Harry Garst 
Ja , en hopelijk kan je er iets mee met de module code :-)
Nog wat verder uitbreiden met nieuwe regels 
groet Jan

@janhardo Testing commando's&nb...

janhardo 935
April 25 2026
0 0

@janhardo 
Testing commando's 
Performance test.comparison, seems to be quick with the module

with(LinearAlgebra):

# matrix size (adjust as needed!)
n := 40:

# random matrices
A := RandomMatrix(n,n, generator=-5..5):
B := RandomMatrix(n,n, generator=-5..5):
C := RandomMatrix(n,n, generator=-5..5):
d := RandomMatrix(n,n, generator=-5..5):

# ---------- 1. Naive method ----------
t1 := time():

K1 := KroneckerProduct(A,B):
K2 := KroneckerProduct(C,d):
R1 := K1 . K2:

t_naive := time() - t1:

# ---------- 2. Smart method (KRON7) ----------
t2 := time():

R2 := KroneckerProduct(A . C, B . d):

t_smart := time() - t2:

# ---------- Results ----------
printf("Naive time  = %.4f sec\n", t_naive);
printf("Smart time  = %.4f sec\n", t_smart);
printf("Speedup(faster)     = %.2f x\n", t_naive / t_smart);

# ---------- Verification ----------
printf("Equal? %a\n", Equal(R1, R2));
Naive time  = 244.7970 sec
Smart time  = 1.2500 sec
Speedup(faster)     = 195.84 x
Equal? true

  > ....

janhardo 935
April 22 2026
0 0
> 

restart;
with(LinearAlgebra):
ZehfussReplace := proc(expr::uneval)
    uses LinearAlgebra:
    local replace;
    replace := proc(det)
        local kr_args, A, B;
        kr_args := op(1, det);
        if not type(kr_args, specfunc(KroneckerProduct)) then return eval(det) end if;
        A := op(1, kr_args);
        B := op(2, kr_args);
        'Determinant'(A)^(RowDimension(B)) * 'Determinant'(B)^(RowDimension(A));
    end proc;
    subsindets(expr, specfunc(specfunc(KroneckerProduct), Determinant), replace);
end proc:

A := Matrix(2,2,symbol=a);
B := Matrix(3,3,symbol=b);

# Important: call the procedure directly with the unevaluated form
result := ZehfussReplace( Determinant(KroneckerProduct(A,B)) + 2*Determinant(KroneckerProduct(A,A)) );
# print(result);
  eval(result);

Matrix(2, 2, {(1, 1) = a[1, 1], (1, 2) = a[1, 2], (2, 1) = a[2, 1], (2, 2) = a[2, 2]})

  

Matrix(%id = 36893490552170632124)

  

LinearAlgebra:-Determinant(A)^3*LinearAlgebra:-Determinant(B)^2+2*LinearAlgebra:-Determinant(A)^4

  

(a[1, 1]*a[2, 2]-a[1, 2]*a[2, 1])^3*(b[1, 1]*b[2, 2]*b[3, 3]-b[1, 1]*b[2, 3]*b[3, 2]-b[1, 2]*b[2, 1]*b[3, 3]+b[1, 2]*b[2, 3]*b[3, 1]+b[1, 3]*b[2, 1]*b[3, 2]-b[1, 3]*b[2, 2]*b[3, 1])^2+2*(a[1, 1]*a[2, 2]-a[1, 2]*a[2, 1])^4

 (1)
 

 

Download Zehfuss_-kronecker_determinant__mprimes_22-4-2026.mw

@Harry Garst  It doesn't s...

janhardo 935
April 21 2026
0 0

@Harry Garst 

It doesn't seem necessary to use two variables.

These are two ways of describing the same variable..

Apparently, the expression hold for all n =1.. 2
proof by induction further

kronecker_-Zehfuss_mprimes_20-4-2026.mw...

janhardo 935
April 20 2026
0 0

kronecker_-Zehfuss_mprimes_20-4-2026.mw

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