# NumPy: Calculate the QR decomposition of a given matrix

## NumPy: Linear Algebra Exercise-13 with Solution

Write a NumPy program to calculate the QR decomposition of a given matrix.

From Wikipedia: In linear algebra, a QR decomposition (also called a QR factorization) of a matrix is a decomposition of a matrix A into a product A = QR of an orthogonal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares problem and is the basis for a particular eigenvalue algorithm, the QR algorithm.

**Square matrix**

Any real square matrix A may be decomposed as

where Q is an orthogonal matrix (its columns are orthogonal unit vectors meaning {\displaystyle Q^{\textsf {T}}Q=QQ^{\textsf {T}}=I} {\displaystyle Q^{\textsf {T}}Q=QQ^{\textsf {T}}=I}) and R is an upper triangular matrix (also called right triangular matrix). If A is invertible, then the factorization is unique if we require the diagonal elements of R to be positive.

If instead A is a complex square matrix, then there is a decomposition A = QR where Q is a unitary matrix (so {\displaystyle Q^{*}Q=QQ^{*}=I} {\displaystyle Q^{*}Q=QQ^{*}=I}).

If A has n linearly independent columns, then the first n columns of Q form an orthonormal basis for the column space of A. More generally, the first k columns of Q form an orthonormal basis for the span of the first k columns of A for any 1 ≤ k ≤ n.[1] The fact that any column k of A only depends on the first k columns of Q is responsible for the triangular form of R.[1]

**Sample Solution** :

**Python Code :**

```
import numpy as np
m = np.array([[1,2],[3,4]])
print("Original matrix:")
print(m)
result = np.linalg.qr(m)
print("Decomposition of the said matrix:")
print(result)
```

Sample Output:

Original matrix: [[1 2] [3 4]] Decomposition of the said matrix: (array([[-0.31622777, -0.9486833 ], [-0.9486833 , 0.31622777]]), array([[-3.16227766, -4.42718872], [ 0. , -0.63245553]]))

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**Previous:** Write a NumPy program to compute the inverse of a given matrix.

**Next:** Write a NumPy program to compute the condition number of a given matrix.

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## Python: Tips of the Day

**Python: Cache results with decorators**

There is a great way to cache functions with decorators in Python. Caching will help save time and precious resources when there is an expensive function at hand.

Implementation is easy, just import lru_cache from functools library and decorate your function using @lru_cache.

from functools import lru_cache @lru_cache(maxsize=None) def fibo(a): if a <= 1: return a else: return fibo(a-1) + fibo(a-2) for i in range(20): print(fibo(i), end="|") print("\n\n", fibo.cache_info())

Output:

0|1|1|2|3|5|8|13|21|34|55|89|144|233|377|610|987|1597|2584|4181| CacheInfo(hits=36, misses=20, maxsize=None, currsize=20)

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