Matrix transformations

Understand the idea

Matrix transformations are coordinate maps. A clockwise n×n rotation maps (r,c) to (c,n-1-r). Transpose then reverse each row implements that map in place. Transpose swaps only positions above the diagonal with their mirrors; the diagonal stays fixed. Reversing rows afterward performs the remaining reflection. Validate shape before mutation so invalid input is rejected without partial transformation.

Work through an example

[[1,2],[3,4]] transposes to [[1,3],[2,4]], then row reversals give [[3,1],[4,2]]. Four rotations restore the original.

Time and space complexity

O(n²) time/O(1) auxiliary state for square in-place rotation.

Why it works

Every source coordinate is moved exactly once to its rotated position through the composed swaps.

Common mistakes

• Reversing columns after transpose rotates the other direction.
• Rectangular rotation changes dimensions and needs another design.

Chapter lab · Python implementation

Arithmetic assumptions affect algorithm correctness. The lab uses Euclid for GCD, modular repeated squaring, and a square-matrix rotation. Python integers avoid fixed-width overflow but grow in storage and arithmetic cost. Treat coordinate conventions and modular inverses explicitly rather than relying on memorized formulas.

This shared lab connects the chapter's topics. Follow the current lesson's worked example first, then trace the complete implementation. Additional algorithms described in topic exercises are variations you can implement independently.

def gcd(a,b):
    a,b=abs(a),abs(b)
    while b: a,b=b,a%b
    return a

def mod_power(base,exponent,modulus):
    if exponent<0 or modulus<1: raise ValueError('invalid exponent/modulus')
    result=1%modulus; base%=modulus
    while exponent:
        if exponent&1: result=result*base%modulus
        base=base*base%modulus; exponent//=2
    return result

def rotate(matrix):
    n=len(matrix)
    if any(len(row)!=n for row in matrix): raise ValueError('square required')
    for r in range(n):
        for c in range(r+1,n):
            matrix[r][c],matrix[c][r]=matrix[c][r],matrix[r][c]
    for row in matrix: row.reverse()

def count_primes(limit):
    if limit<2: return 0
    prime=bytearray([1])*limit
    prime[0]=prime[1]=0
    p=2
    while p*p<limit:
        if prime[p]:
            for multiple in range(p*p,limit,p):
                prime[multiple]=0
        p+=1
    return sum(prime)

def digit_power(base,digits,modulus):
    if modulus<1 or any(not 0<=d<=9 for d in digits):
        raise ValueError('positive modulus and decimal digits required')
    result=1%modulus
    for digit in digits:
        result=mod_power(result,10,modulus)*mod_power(base,digit,modulus)%modulus
    return result

Run and test the lab

Copy the implementation and these assertions into a Python 3 file and run it. An assertion failure means the actual result differs from the expectation. Add the current topic's exercise and boundary tests to deepen your understanding.

assert gcd(48,18)==6 and gcd(-8,0)==8 and gcd(0,0)==0
assert mod_power(3,5,7)==pow(3,5,7)==5
assert mod_power(9,0,1)==0
a=[[1,2],[3,4]]; rotate(a); assert a==[[3,1],[4,2]]
assert all(mod_power(5,e,13)==pow(5,e,13) for e in range(30))
assert count_primes(10)==4 and count_primes(0)==0
assert count_primes(2)==0 and count_primes(100)==25
assert digit_power(2,[1,0],1337)==pow(2,10,1337)
assert digit_power(3,[2,0,1],1337)==pow(3,201,1337)