Numpy & Scipy - 1.3 Basic Manipulation of Matrices (1)

copy (deep copy)

• np.copy: Deep copy, allocates new memory space by referencing.

a = np.array([[1, 2.5, 3], [-1, -2, -1.5], [4, 5.5, 6]], dtype=np.float64)

b = a # Shallow copy, references the same memory space.

a[0,0] = 0.0 # This changes both a[0,0] and b[0,0] to 0.

b = np.copy(a) # Deep copy, creates a matrix with the same values but in a completely separate memory space.

prt(a, fmt="%0.2f")

reshape (swallow copy)

• np.reshape(matrix, number or shape): Changes the input matrix into the desired 1D vector or 2D matrix shape and performs a shallow copy.
• Here, shape is a tuple type. e.g., (2,3) means a 2x3 matrix.
• If you input a single number, it becomes a 1D vector like (3,).

a = np.array([[1, 2.5, 3], [-1, -2, -1.5], [4, 5.5, 6]], dtype=np.float64) 
# 2x3, total 6 entries.

b = np.reshape(a, 6) # Becomes a 1D array (vector).

print(b)

b = np.reshape(a, (3,2)) # Becomes a 2D array (matrix). 3x2 = 6, so entry count matches.
# Error occurs if entry count differs.

print(b)

# If you don't want a shallow copy,
# use copy to perform a deep copy.
b = np.copy(np.reshape(a, (3,2))) 

[1, 2.5, 3, -1, -2, -1.5]

[[1, 2.5]
[3, -1]
[-2, -1.5]]

tril / triu

• np.tril(matrix, band_id): Deep copies the lower part including band_id. Lower triangular matrix.
• band_id can be omitted as a default parameter.

a = np.array([[1,2,3], [4,5,6], [7,8,9]], dtype=np.float64)

b = np.tril(a)

prt(b, "%0.2f", delimiter=" , ")

 1.00 ,  0.00 ,  0.00
 4.00 ,  5.00 ,  0.00
 7.00 ,  8.00 ,  9.00

• np.triu(matrix, band_id): Deep copies the upper part including band_id. Upper triangular matrix.

a = np.array([[1,2,3], [4,5,6], [7,8,9]], dtype=np.float64)

b = np.triu(a)

prt(b, "%0.2f", delimiter=" , ")

 1.00 ,  2.00 ,  3.00
 0.00 ,  5.00 ,  6.00
 0.00 ,  0.00 ,  9.00

diag

• np.diag(matrix, k=band_id): Function to extract a specific band and create a 1D array (vector) via shallow copy.

a = np.array([[1,2,3], [4,5,6], [7,8,9]], dtype=np.float64)

b = np.diag(a, k=1)

prt(b, "%0.2f", delimiter=" , ")

b[0] = 0.0 # Error occurs because it is readonly

 2.00 ,  6.00

np.diag behavior changes with input

• If you input a 1D array into matrix in np.diag(matrix, k=band_id), it returns a deep copied square matrix.

c = np.array([1,2,3,4] , dtype=np.float64) # When a 1D array is input

d = np.diag(c, k=-1) # It is a deep copy.

c[0] = 0.0

prt(d, "%0.2f", delimiter=" , ")

 0.00 ,  0.00 ,  0.00 ,  0.00 ,  0.00
 1.00 ,  0.00 ,  0.00 ,  0.00 ,  0.00
 0.00 ,  2.00 ,  0.00 ,  0.00 ,  0.00
 0.00 ,  0.00 ,  3.00 ,  0.00 ,  0.00
 0.00 ,  0.00 ,  0.00 ,  4.00 ,  0.00

diagflat

• np.diagflat(M, k=band_id): Always creates a deep copied square matrix.
• Note that it flattens the input M into a 1D array before creating the square matrix.
• 1d array -> similar to np.diag, 2d array -> flattens to 1d then creates square matrix.

M = np.array([[1,3],[2,4]], dtype=np.float64)

e = np.diagflat(M, k = 0)

prt(e, "%0.2f", delimiter=" , ")

 1.00 ,  0.00 ,  0.00 ,  0.00
 0.00 ,  3.00 ,  0.00 ,  0.00
 0.00 ,  0.00 ,  2.00 ,  0.00
 0.00 ,  0.00 ,  0.00 ,  4.00

trace

• np.trace: Returns the sum of diagonal entries or band entries.

T = np.array([[1,2,3],[4,5,6],[7,8,9]], dtype=np.float64)

val = np.trace(T)

print(val)

val = np.trace(T, offset=-1)  # uses offset, not k

print(val)

15

12

flatten method / ravel

• flatten: Flattens matrix A into a 1D array, deep copies, and returns it.

T = np.array([[1,2,3],[4,5,6],[7,8,9]], dtype=np.float64)

flat = T.flatten()

prt(flat, "%0.2f", delimiter=" , ")

1.00 ,  2.00 ,  3.00 ,  4.00 ,  5.00 ,  6.00 ,  7.00 ,  8.00 ,  9.00

• np.ravel: Flattens matrix A into a 1D array, shallow copies, and returns it.

T = np.array([[1,2,3],[4,5,6],[7,8,9]], dtype=np.float64)

rav = np.ravel(T)

T[0,0] = 0.0

prt(rav, "%0.2f", delimiter=" , ")

 0.00 ,  2.00 ,  3.00 ,  4.00 ,  5.00 ,  6.00 ,  7.00 ,  8.00 ,  9.00