Matrix Decomposition¶

The idea of Matrix decomposition also known as matrix factorization

Matrix decompositions are an important step in solving linear systems in a computationally efficient manner
Numerous decomposition exist examples include: Cholesky Decomposition, LU Decomposition, QR decompositon and Eigendecomposition

Eigendecomposition¶

Let \(A\) be an \(n \times n\) matrix and \(\mathbf{x}\) be an \(n \times 1\) nonzero vector. An eigenvalue of \(A\) is a number \(\lambda\) such that

\[A \boldsymbol{x} = \lambda \boldsymbol{x}\]

A vector \(\mathbf{x}\) satisfying this equation is called an eigenvector associated with \(\lambda\)

>>> a = np.diag((1, 2, 3))
>>> a
array([[1, 0, 0],
       [0, 2, 0],
       [0, 0, 3]])
>>> w,v = np.linalg.eig(a)
>>> w;v
array([ 1.,  2.,  3.])
array([[ 1.,  0.,  0.],
       [ 0.,  1.,  0.],
       [ 0.,  0.,  1.]])

Eigenvectors and eigenvalues are important mathematical identities that play many roles across a range of disciplines