matrix operation Algorithm
Algorithm that are tailored to particular matrix structures, such as sparse matrix and near-diagonal matrix, expedite calculations in finite component method and other calculations. In mathematics, a matrix (plural matrix) is a rectangular array (see irregular matrix) of numbers, symbols, or expressions, arranged in rows and columns.
The term" matrix" (Latin for" womb", derived from mater — mother) was coined by James Joseph Sylvester in 1850, who understand a matrix as an object give rise to a number of determinants today named minors, that is to say, determinants of smaller matrix that deduce from the original one by remove columns and rows. The inception of matrix mechanics by Heisenberg, birth and Jordan led to analyzing matrix with infinitely many rows and columns. Many theorems were first established for small matrix only, for example the Cayley – Hamilton theorem was proved for 2×2 matrix by Cayley in the aforementioned memoir, and by Hamilton for 4×4 matrix.
"""
function based version of matrix operations, which are just 2D arrays
"""
def add(matrix_a, matrix_b):
if _check_not_integer(matrix_a) and _check_not_integer(matrix_b):
rows, cols = _verify_matrix_sizes(matrix_a, matrix_b)
matrix_c = []
for i in range(rows[0]):
list_1 = []
for j in range(cols[0]):
val = matrix_a[i][j] + matrix_b[i][j]
list_1.append(val)
matrix_c.append(list_1)
return matrix_c
def subtract(matrix_a, matrix_b):
if _check_not_integer(matrix_a) and _check_not_integer(matrix_b):
rows, cols = _verify_matrix_sizes(matrix_a, matrix_b)
matrix_c = []
for i in range(rows[0]):
list_1 = []
for j in range(cols[0]):
val = matrix_a[i][j] - matrix_b[i][j]
list_1.append(val)
matrix_c.append(list_1)
return matrix_c
def scalar_multiply(matrix, n):
return [[x * n for x in row] for row in matrix]
def multiply(matrix_a, matrix_b):
if _check_not_integer(matrix_a) and _check_not_integer(matrix_b):
matrix_c = []
rows, cols = _verify_matrix_sizes(matrix_a, matrix_b)
if cols[0] != rows[1]:
raise ValueError(
f"Cannot multiply matrix of dimensions ({rows[0]},{cols[0]}) "
f"and ({rows[1]},{cols[1]})"
)
for i in range(rows[0]):
list_1 = []
for j in range(cols[1]):
val = 0
for k in range(cols[1]):
val = val + matrix_a[i][k] * matrix_b[k][j]
list_1.append(val)
matrix_c.append(list_1)
return matrix_c
def identity(n):
"""
:param n: dimension for nxn matrix
:type n: int
:return: Identity matrix of shape [n, n]
"""
n = int(n)
return [[int(row == column) for column in range(n)] for row in range(n)]
def transpose(matrix, return_map=True):
if _check_not_integer(matrix):
if return_map:
return map(list, zip(*matrix))
else:
# mt = []
# for i in range(len(matrix[0])):
# mt.append([row[i] for row in matrix])
# return mt
return [[row[i] for row in matrix] for i in range(len(matrix[0]))]
def minor(matrix, row, column):
minor = matrix[:row] + matrix[row + 1 :]
minor = [row[:column] + row[column + 1 :] for row in minor]
return minor
def determinant(matrix):
if len(matrix) == 1:
return matrix[0][0]
res = 0
for x in range(len(matrix)):
res += matrix[0][x] * determinant(minor(matrix, 0, x)) * (-1) ** x
return res
def inverse(matrix):
det = determinant(matrix)
if det == 0:
return None
matrix_minor = [[] for _ in range(len(matrix))]
for i in range(len(matrix)):
for j in range(len(matrix)):
matrix_minor[i].append(determinant(minor(matrix, i, j)))
cofactors = [
[x * (-1) ** (row + col) for col, x in enumerate(matrix_minor[row])]
for row in range(len(matrix))
]
adjugate = transpose(cofactors)
return scalar_multiply(adjugate, 1 / det)
def _check_not_integer(matrix):
if not isinstance(matrix, int) and not isinstance(matrix[0], int):
return True
raise TypeError("Expected a matrix, got int/list instead")
def _shape(matrix):
return list((len(matrix), len(matrix[0])))
def _verify_matrix_sizes(matrix_a, matrix_b):
shape = _shape(matrix_a)
shape += _shape(matrix_b)
if shape[0] != shape[2] or shape[1] != shape[3]:
raise ValueError(
f"operands could not be broadcast together with shape "
f"({shape[0], shape[1]}), ({shape[2], shape[3]})"
)
return [shape[0], shape[2]], [shape[1], shape[3]]
def main():
matrix_a = [[12, 10], [3, 9]]
matrix_b = [[3, 4], [7, 4]]
matrix_c = [[11, 12, 13, 14], [21, 22, 23, 24], [31, 32, 33, 34], [41, 42, 43, 44]]
matrix_d = [[3, 0, 2], [2, 0, -2], [0, 1, 1]]
print(
"Add Operation, %s + %s = %s \n"
% (matrix_a, matrix_b, (add(matrix_a, matrix_b)))
)
print(
"Multiply Operation, %s * %s = %s \n"
% (matrix_a, matrix_b, multiply(matrix_a, matrix_b))
)
print("Identity: %s \n" % identity(5))
print("Minor of {} = {} \n".format(matrix_c, minor(matrix_c, 1, 2)))
print("Determinant of {} = {} \n".format(matrix_b, determinant(matrix_b)))
print("Inverse of {} = {}\n".format(matrix_d, inverse(matrix_d)))
if __name__ == "__main__":
main()
