#!/usr/bin/env python
# coding: utf-8

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#imports 
import numpy as np
from numpy import linalg


# In this tutorial, we will make use of numpy's ```linalg``` module. 

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# first, initialize new numpy array 
a = np.array([[1],[2],[3]])
a


# ---
# We can compute various norms using the ```linalg.norm``` function:

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linalg.norm(a)  # L2 / Euclidean norm


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linalg.norm(a, ord=1)  # L1 norm


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linalg.norm(a, ord=np.inf)  # inf-norm / max norm


# ---
# Computing the Determinant:

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# initialize matrix A
A = np.array([[1,0,3],[4,-1,6],[7,8,3]])
A


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# compute determinant
linalg.det(A)


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# initialize matrix B
B = np.array([[1,2,3],[2,4,6],[3,6,9]])
B


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# compute determinant
linalg.det(B)


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# compute inverse
try:
    linalg.inv(B)  # throws exception because matrix is singular 
except Exception as e:
    print(e)


# ---
# Computing eigenvalues and eigenvectors:

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eigvals, eigvecs = linalg.eig(A)  


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print(eigvals) # vector of eigenvalues


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print(eigvecs) # eigenvectors are columns of the matrix


# Note that ```eig``` does not guarantee that eigenvalues are returned in sorted order.
# 
# For symmetric matrices, use ```eigh``` instead. This is more efficient and numerically reliable because it takes advantage of the fact that the matrix is symmetric. Moreover, the eigenvalues are returned in ascending order.

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# initialize symmetric matrix
S = np.matmul(A, A.transpose())
S


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eigvals, eigvecs = linalg.eigh(S)   


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print(eigvals)


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print(eigvecs)


# If you only want eigenvalues, you can use the ```linalg.eigvals``` and ```linalg.eigvalsh``` functions.

# ---
# Let's try to recompose the matrix from eigenvalues and eigenvectors:

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A


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L, V = linalg.eig(A)  # eigenvalues and eigenvectors


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A_recomposed = np.matmul(V, np.matmul(np.diag(L), linalg.inv(V)))  # recompose matrix 
A_recomposed


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