Full and Identity¶

NumPy provides functions for creating constant-filled arrays and identity matrices commonly used in linear algebra.

np.full Function¶

Creates an array filled with a specified constant value.

1. Basic Usage¶

```python import numpy as np

def main(): a = np.full((2, 5), 7) print("np.full((2, 5), 7)") print(a)

if name == "main": main() ```

Output:

np.full((2, 5), 7) [[7 7 7 7 7] [7 7 7 7 7]]

2. Any Fill Value¶

The fill value can be any scalar: integer, float, or complex.

np.eye Function¶

Creates a 2D array with ones on the diagonal and zeros elsewhere.

1. Square Matrix¶

```python import numpy as np

def main(): a = np.eye(3) print("np.eye(3)") print(a)

if name == "main": main() ```

Output:

np.eye(3) [[1. 0. 0.] [0. 1. 0.] [0. 0. 1.]]

2. Rectangular Matrix¶

```python import numpy as np

def main(): a = np.eye(3, 5) print("np.eye(3, 5)") print(a)

if name == "main": main() ```

Output:

np.eye(3, 5) [[1. 0. 0. 0. 0.] [0. 1. 0. 0. 0.] [0. 0. 1. 0. 0.]]

np.identity Function¶

Creates a square identity matrix.

1. Basic Usage¶

```python import numpy as np

def main(): a = np.identity(3) print("np.identity(3)") print(a)

if name == "main": main() ```

Output:

np.identity(3) [[1. 0. 0.] [0. 1. 0.] [0. 0. 1.]]

2. Square Only¶

```python import numpy as np

def main(): try: a = np.identity(3, 5) except TypeError as e: print(e) print("np.identity only creates square matrices.")

if name == "main": main() ```

Use np.eye for non-square matrices with diagonal ones.

eye vs identity¶

Both create identity-like matrices but differ in flexibility.

1. np.eye Flexibility¶

np.eye(N, M) accepts two shape parameters for rectangular output.

2. np.identity Simplicity¶

np.identity(N) only accepts one parameter, always producing square matrices.

3. Recommendation¶

Prefer np.eye for its greater flexibility in all cases.

Linear Algebra Use¶

Identity matrices are fundamental in matrix operations.

1. Matrix Inverse¶

\(A \cdot A^{-1} = I\) where \(I\) is the identity matrix.

2. Eigenvalue Problems¶

\(A \cdot v = \lambda \cdot v\) can be rewritten as \((A - \lambda I) \cdot v = 0\).

3. Basis Vectors¶

The columns of an identity matrix form the standard basis vectors.

Exercises¶

Exercise 1. Create a 4x4 matrix filled with the value np.pi using np.full. Print the matrix and confirm that every element equals np.pi.

import numpy as np

M = np.full((4, 4), np.pi)
print(M)
print(f"All pi: {np.all(M == np.pi)}")

Exercise 2. Create a 3x5 rectangular identity-like matrix using np.eye(3, 5). Then verify that multiplying a (3, 5) matrix by this identity-like matrix (transposed to (5, 3)) and then by the original produces a (3, 3) result. Print the shapes at each step.

import numpy as np

E = np.eye(3, 5)
A = np.random.randn(3, 5)
result = A @ E.T
print(f"E shape: {E.shape}")         # (3, 5)
print(f"A shape: {A.shape}")         # (3, 5)
print(f"A @ E.T shape: {result.shape}") # (3, 3)

Exercise 3. Using np.eye with the k parameter, create a 5x5 matrix with ones on the first superdiagonal (k=1). Then create another with ones on the first subdiagonal (k=-1). Add both to the standard identity matrix np.eye(5) to form a tridiagonal matrix of ones. Print the result.

import numpy as np

I = np.eye(5)
upper = np.eye(5, k=1)
lower = np.eye(5, k=-1)
tridiag = I + upper + lower
print(tridiag)