Linear Algebra

Linear algebra works with vectors and matrices. NumPy has the products at the top level and the matrix tools in np.linalg. Results are floats, so compare them with np.allclose instead of ==.

dot, inner, and outer

For two 1-D arrays, dot and inner multiply matching values and add them up. For 2-D arrays, inner pairs each row of the first with each row of the second. outer multiplies every value of one array by every value of the other.

import numpy as np

u = np.array([1, 3, 5])
w = np.array([2, 4, 6])
print(np.dot(u, w), np.inner(u, w))
print(np.outer(u, w))

a = np.array([[1, 3], [5, 7]])
b = np.array([[2, 4], [6, 8]])
print(np.inner(a, b))

Output:

44 44
[[ 2  4  6]
 [ 6 12 18]
 [10 20 30]]
[[14 30]
 [38 86]]

The dot product is 1×2 + 3×4 + 5×6 = 44. In the last result, the top-left value is row 0 of a with row 0 of b: 1×2 + 3×4 = 14.

Transpose and trace

.T swaps rows and columns. np.trace adds up the main diagonal.

import numpy as np

m = np.array([[6, 3, 5], [9, 2, 1], [7, 8, 4]])
print(m.T)
print(np.trace(m))

Output:

[[6 9 7]
 [3 2 8]
 [5 1 4]]
12

Determinant and inverse

np.linalg.det gives the determinant of a square matrix. When it is not 0, np.linalg.inv gives the inverse, and the matrix times its inverse is the identity. A matrix with determinant 0 has no inverse and raises LinAlgError.

import numpy as np

A = np.array([[2, 4], [6, 8]])
print(round(np.linalg.det(A), 6))
inverse = np.linalg.inv(A)
print(inverse)
print(np.allclose(A @ inverse, np.eye(2)))

try:
    np.linalg.inv(np.array([[1, 2], [2, 4]]))
except np.linalg.LinAlgError as error:
    print("LinAlgError:", error)

Output:

-8.0
[[-1.    0.5 ]
 [ 0.75 -0.25]]
True
LinAlgError: Singular matrix

Solve a system of equations

np.linalg.solve(A, b) finds x with A @ x == b. Here it solves 2x + 4y = 5 and 6x + 8y = 6. A must be square with a non-zero determinant.

import numpy as np

A = np.array([[2, 4], [6, 8]])
b = np.array([5, 6])
x = np.linalg.solve(A, b)
print(x)
print(np.allclose(A @ x, b))

Output:

[-2.    2.25]
True

Key takeaways

• dot and inner agree for 1-D. outer pairs every value

• .T transposes and trace adds the diagonal

• inv and solve need a square matrix with a non-zero determinant

• Check float results with np.allclose

• Next: Saving and Loading Arrays

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