Exercise 1.1.24

Question

Create a $3$ by $2$ matrix $A$ with rank $2$. Factor $A$ into $A=C M R$.
The reason for this page is that the factorizations $A=C R$ and $A=C M R$ have jumped forward in importance for large matrices. When $C$ takes columns directly from $A$, and $\boldsymbol{R}$ takes rows directly from $A$, those matrices preserve properties that are lost in the more famous $Q R$ and SVD factorizations. Where $A=Q R$ and $A=U \Sigma V^{\mathrm{T}}$ involve orthogonalizing the vectors, $C$ and $R$ keep the original data:
\begin{itemize}
\item If $A$ is nonnegative, so are $C$ and $R$.
\item If $A$ is sparse, so are $C$ and $R$.
\end{itemize}

Neil Z. · Accepted Answer

We pick $A=\begin{bmatrix}1 &3\2&5\3&6\end{bmatrix}$, then we have
$$A=CR=\begin{bmatrix}1 &3\2&5\3&6\end{bmatrix}I.$$

Pick $R=\begin{bmatrix}1&3\2&5\end{bmatrix}$, so we have
$$M= \begin{bmatrix}-8&11\-15&23\end{bmatrix}.$$

\begin{lstlisting}[language=python]
import numpy as np

R = np.array([[1, 3], [2,5]])#.reshape(1,2)
A = np.array([[1,3],[2,5],[3,6]])
C = A

a = np.matmul(C.transpose(), A)
b = np.matmul(a, R.transpose())
k = np.matmul(C.transpose(), C)
invc = np.linalg.inv(k)
invr = np.linalg.inv(np.matmul(R, R.transpose()))
m = np.matmul(a, b)
f = np.matmul(invc, m)
f = np.matmul(f, invr)
invc, invr, m, f
\end{lstlisting}
\begin{verbatim}
(array([[ 3.68421053, -1.63157895],
        [-1.63157895,  0.73684211]]), array([[ 29., -17.],
        [-17.,  10.]]), array([[ 8969, 15334],
        [20187, 34513]]), array([[ -8.,  11.],
        [-15.,  23.]]))
\end{verbatim}

Exercise 1.1.24

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Exercise 1.1.24

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