Implementing Matrix Decomposition in Python
Matrix Decomposition Techniques are essential tools in numerical linear algebra, powering solutions for systems of equations, stability analysis, and matrix inversion.
Setting Up the Matrix
We define a 2×2 matrix A to use for both decompositions:
# Define a 2x2 matrix A
A = np.array([[4, 3],
[6, 3]])
This matrix will be decomposed into lower/upper and orthogonal/triangular components.
Performing LU Decomposition
LU decomposition splits a matrix into:
L: lower triangular;U: upper triangular;P: permutation matrix to account for row swaps.
123456789101112import numpy as np from scipy.linalg import lu # Define a 2x2 matrix A A = np.array([[4, 3], [6, 3]]) # Perform LU decomposition: P, L, U such that P @ A = L @ U P, L, U = lu(A) # Verify that P @ A equals L @ U by reconstructing A from L and U print(np.dot(L, U))
Why this matters: LU decomposition is heavily used in numerical methods to solve linear systems and invert matrices efficiently.
Performing QR Decomposition
QR decomposition factors a matrix into:
Q: Orthogonal matrix (preserves angles/lengths)R: Upper triangular matrix
# Perform QR decomposition: Q (orthogonal), R (upper triangular)
Q, R = qr(A)
The original matrix is then:
# Verify that Q @ R equals A by reconstructing A from Q and R
print(np.dot(Q, R))
123456789101112import numpy as np from scipy.linalg import qr # Define a 2x2 matrix A A = np.array([[4, 3], [6, 3]]) # Perform QR decomposition: Q (orthogonal), R (upper triangular) Q, R = qr(A) # Verify that Q @ R equals A by reconstructing A from Q and R print(np.dot(Q, R))
Why this matters: QR is commonly used for solving least squares problems and is more numerically stable than LU in some scenarios.
1. What is the role of the permutation matrix P in LU decomposition?
2. What shape is the matrix U in an LU decomposition?
3. In QR decomposition, what property does the matrix Q have?
4. Which of the following verifies the LU decomposition correctly?
5. Suppose you need to solve the system A⋅x=b using QR decomposition. What code adjustment would you need to make?
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Can you explain the difference between LU and QR decomposition in more detail?
What are some practical applications of these decompositions?
Can you show how to use these decompositions to solve a system of equations?
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Matrix Decomposition Techniques are essential tools in numerical linear algebra, powering solutions for systems of equations, stability analysis, and matrix inversion.
Setting Up the Matrix
We define a 2×2 matrix A to use for both decompositions:
# Define a 2x2 matrix A
A = np.array([[4, 3],
[6, 3]])
This matrix will be decomposed into lower/upper and orthogonal/triangular components.
Performing LU Decomposition
LU decomposition splits a matrix into:
L: lower triangular;U: upper triangular;P: permutation matrix to account for row swaps.
123456789101112import numpy as np from scipy.linalg import lu # Define a 2x2 matrix A A = np.array([[4, 3], [6, 3]]) # Perform LU decomposition: P, L, U such that P @ A = L @ U P, L, U = lu(A) # Verify that P @ A equals L @ U by reconstructing A from L and U print(np.dot(L, U))
Why this matters: LU decomposition is heavily used in numerical methods to solve linear systems and invert matrices efficiently.
Performing QR Decomposition
QR decomposition factors a matrix into:
Q: Orthogonal matrix (preserves angles/lengths)R: Upper triangular matrix
# Perform QR decomposition: Q (orthogonal), R (upper triangular)
Q, R = qr(A)
The original matrix is then:
# Verify that Q @ R equals A by reconstructing A from Q and R
print(np.dot(Q, R))
123456789101112import numpy as np from scipy.linalg import qr # Define a 2x2 matrix A A = np.array([[4, 3], [6, 3]]) # Perform QR decomposition: Q (orthogonal), R (upper triangular) Q, R = qr(A) # Verify that Q @ R equals A by reconstructing A from Q and R print(np.dot(Q, R))
Why this matters: QR is commonly used for solving least squares problems and is more numerically stable than LU in some scenarios.
1. What is the role of the permutation matrix P in LU decomposition?
2. What shape is the matrix U in an LU decomposition?
3. In QR decomposition, what property does the matrix Q have?
4. Which of the following verifies the LU decomposition correctly?
5. Suppose you need to solve the system A⋅x=b using QR decomposition. What code adjustment would you need to make?
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