fix bug where triang would give a diagonalisable matrix

README.md

@@ -1,2 +1,23 @@
 # LinAlg-Practice
 TUI program to practice linear algebra computations (multiplications, determinants, inverses, eigenvalues, etc). Written in python by leveraging the sympy library.
+
+## Usage
+To use: run the program with your favourite python interpreter
+```sh
+python linalg-practice.py
+```
+
+You will be prompted for selecting which settings you want to use
+
+## Features
+* Multiplication of two matrices
+* Determinant computation
+* Inverse computation
+* Eigenvalue computation
+* Finding diagonalisations
+* Finding triangularisations
+
+## Planned:
+* Linear systems
+* Row-echelon form
+* GUI??

linalg-practice.py

@@ -1,6 +1,18 @@
 import sympy as sp
 import time
+import datetime
 import random
+import readline
+
+errs = (ValueError, TypeError)
+
+def genmatrix(rowcol, intmax, dif) :
+    # generate a random matrices
+    a = sp.randMatrix(rowcol, rowcol, -intmax, intmax)
+    # if determinant non-zero, radnom value less than difficulty, set a to its inverse
+    if a.det() != 0 and random.random() <= dif :
+        a = a.inv()
+    return a
 
 # in function to take arbitrary matrix input from user
 def inmat(rowcol) :
@@ -12,7 +24,11 @@ def inmat(rowcol) :
         mat[i]=input().split(" ")
         for j in range(len(mat[i])) :
             # convert list using nsimplify in order to take rational number symbolically
+            try :
                 mat[i][j] = sp.nsimplify(mat[i][j])
+            except errs : return False
+    try : mat = sp.Matrix(mat)
+    except errs : return False
     return mat
 
 # multcheck function to check if two matrices are multiplied together
@@ -20,38 +36,66 @@ def multcheck(a, b, rowcol, intmax) :
     sp.pprint(b)
     print("Multiply these two matrices together")
     # return bool based on if input equals the two matrices multiplied together
-    return (sp.Matrix(inmat(rowcol)) == a*b)
+    return (inmat(rowcol) == a*b)
 
 # detcheck function to check if the determinant of the matrix is correct
 def detcheck(a, b, rowcol, intmax):
     det = sp.det(a)
     # return bool based on if input equals determinant
-    return (det == sp.nsimplify(input("What is the determinant of this matrix?: ")))
+    try : d = sp.nsimplify(input("What is the determinant of this matrix?: "))
+    except errs : return False
+    return (det == d)
 
 # invcheck function to check if the inverse of the matrix is correct
 def invcheck(a, b, rowcol, intmax):
-    det = a.det()
-    if det != 0 :
     # return bool based on if input equals inverse of matrix
     print("What is the inverse of this matrix?")
-        return (sp.Matrix(inmat(rowcol)) == a.inv())
+    return (inmat(rowcol) == a.inv())
 
 # eigcheck function to check if the eigenvalues are correct
 def eigcheck(a, b, rowcol, intmax):
     eigs = a.eigenvals()
     for i in range(len(eigs)) :
+        try :
             val = sp.nsimplify(input("Input eigenvalue: "))
-        if not (val in eigs and eigs[val] == sp.nsimplify(input("Input its algebraic multiplicity: "))) :
+            algm = sp.nsimplify(input("Input its algebraic multiplicity: "))
+        except errs : return False
+        if not (val in eigs and eigs[val] == algm) :
             # return false if the eigenvalue not in dictionary and wrong alg multiplicity
             return False
     return True
 
+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
-    rowcol = int(input("What size of matrix do you want to practice with? "))
-    intmax = int(input("What maximum size of integer do you want the matrix to be made out of? "))
-    dif = float(input("What difficulty (probability for a matrix of rational values between 0, 1) do you want? "))
 
     # choose the function of the program that you want
     if t == "mult" :
@@ -62,24 +106,56 @@ def practice(t) :
         f = invcheck
     elif t == "eig" :
         f = eigcheck
+    elif t == "diag" :
+        f = diagcheck
+    elif t == "triang" :
+        f = triangcheck
     else :
         exit()
 
+    while True :
+        try :
+            rowcol = abs(int(input("What size of matrix do you want to practice with? ")))
+            intmax = abs(int(input("What maximum size of integer do you want the matrix to be made out of? ")))
+            dif = float(input("What difficulty (probability for a matrix of rational values between 0, 1) do you want? "))
+            break
+        except errs:
+            continue
+
     # initialise time measurement
     tic = time.perf_counter()
 
     # infinite loop of practice
     while True :
-        # generate a random matrices
+        a = genmatrix(rowcol, intmax, dif)
-        a = sp.randMatrix(rowcol, rowcol, 0, intmax)
+        b = genmatrix(rowcol, intmax, dif)
-        b = sp.randMatrix(rowcol, rowcol, 0, intmax)
+        if t == "triang" :
-
+            while True :
-        # if determinant non-zero, radnom value less than difficulty, set a to its inverse
+                # generate matrices with smaller integers
-        if a.det() != 0 and random.random() <= dif :
+                a = genmatrix(rowcol, intmax//rowcol, dif)
-            a = a.inv()
+                # 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 or 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 :
+            # if diagcheck, make sure the matrix is diagonalizable
+            if t == "diag" :
+                while not a.is_diagonalizable :
+                    a = genmatrix(rowcol, intmax, dif)
+            if t == "inv" :
+                while a.det() == 0 :
+                    a = genmatrix(rowcol, intmax, dif)
             sp.pprint(a)
             # if return of function is True
             if f(a, b, rowcol, intmax) :
@@ -89,7 +165,7 @@ def practice(t) :
                 # increment count
                 count += 1
                 # print success message
-                print("Correct! You have solved", count, "problems in %.2f seconds" %tictoc)
+                print("Correct! You have solved", count, "problems in", str(datetime.timedelta(seconds=round(tictoc))))
                 break
             else :
                 # print failure message
@@ -97,5 +173,5 @@ 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) ")
+t = input("What do you want to practice? (mult, det, inv, eig, diag, triang) ")
 practice(t)