implement triangularisation (triangcheck) feature

linalg-practice.py

@@ -67,6 +67,31 @@ def eigcheck(a, b, rowcol, intmax):
 def diagcheck(a, b, rowcol, intmax):
     return (a.diagonalize()[1] == inmat(rowcol))
 
+# pascalmat function to generate pascal matrices (used for generating unimodular matrices)
+def pascalmat(n) :
+    mat = []
+    # generate rows of binomial coefficients
+    for i in range(n) :
+        mat.append(sp.binomial_coefficients_list(i) + [0]*(n-1-i))
+    mat = sp.Matrix(mat)
+    # make a lower triangular pascal matrix
+    lmat = mat.transpose()
+    rand = random.random()
+    # randomly pick either lower or upper triangular pascal matrix
+    if rand <= 1/2 :
+        mat = lmat
+    return mat
+
+def triangcheck(a, b, rowcol, intmax) :
+    print("Input triangularised matrix,")
+    T = inmat(rowcol)
+    print("Input basis change matrix (from standard basis)")
+    P_inv = inmat(rowcol)
+    # check if P_inv is invertible
+    if P_inv.det() == 0 : return False
+    # return false if T is not upper triangular and not similar to the original matrix
+    return (T.is_upper and a == P_inv.inv()*T*P_inv)
+
 # practice function
 def practice(t) :
     count = 0
@@ -82,6 +107,8 @@ def practice(t) :
         f = eigcheck
     elif t == "diag" :
         f = diagcheck
+    elif t == "triang" :
+        f = triangcheck
     else :
         exit()
 
@@ -101,6 +128,23 @@ def practice(t) :
     while True :
         a = genmatrix(rowcol, intmax, dif)
         b = genmatrix(rowcol, intmax, dif)
+        if t == "triang" :
+            while True :
+                # generate matrices with smaller integers
+                a = genmatrix(rowcol, intmax//rowcol, dif)
+                # ensure a is invertible
+                if a.det() == 0 : continue
+                # find a non-fractional upper trianagular matrix (ensures triangularisability)
+                c = a.LUdecompositionFF()[3]
+                croots = {k: k for k in sp.roots(c.charpoly()) if k not in sp.roots(c.charpoly(), filter='Z')}
+                # make sure only integer roots and c is not diagonalisable
+                if not (len(croots) != 0 and c.is_diagonalizable) :
+                    a = c
+                    pmat = pascalmat(rowcol)
+                    # multiply by a pascal matrix to get a similar matrix with nice numbers
+                    a = pmat.inv()*a*pmat
+                    if a.is_upper: continue
+                    break
 
         # infinite loop until user succeeds 
         while True :
@@ -128,5 +172,5 @@ def practice(t) :
                 continue
 
 # take input from user on the type of practice they want to do
-t = input("What do you want to practice? (mult, det, inv, eig, diag) ")
+t = input("What do you want to practice? (mult, det, inv, eig, diag, triang) ")
 practice(t)