In order to successfully complete this assignment, you must do the required reading, watch the provided videos, and complete all instructions. The embedded survey form must be entirely filled out and submitted on or before 11:59pm on the day before class. Students must come to class the next day prepared to discuss the material covered in this assignment.

Pre-Class Assignment: Decompositions¶

1. Matrix Decomposition¶

✅ **DO THIS:** Watch the following video and answer the questions below.

from IPython.display import YouTubeVideo
YouTubeVideo("-_2he4J6Xxw",width=640,height=360, cc_load_policy=True)

Consider the following code to calculate the $A = Q\Lambda Q^{-1}$ eivendecomposition.

%matplotlib inline
import matplotlib.pylab as plt
import numpy as np
import sympy as sym
sym.init_printing(use_unicode=True)

# Here is our input matrix
A = np.matrix([[15,7,-7],[-1,1,1],[13,7,-5]])
sym.Matrix(A)

# Calculate eigenvalues and vectors using Numpy
e, Q = np.linalg.eig(A)
print(e)
sym.Matrix(Q)

[8. 1. 2.]

#Turn eigenvalues into a diagonal matrix  (there is even a function for that!)
L = np.diag(e)
sym.Matrix(L)

# Calculate A again from Q and L

A2 = Q*L*np.linalg.inv(Q)

sym.Matrix(A2)

✅ **DO THIS:** Using code, verify that A2 is the same as $A$.

# Put your answer here

✅ **DO THIS:** Turn the above code into a function called eigendecomp which takes in a matrix A and returns Q and L.

# Put your code here

✅ **QUESTION:** What other decompositions have we covered in the class so far? Make a list and write down a short description on why we use each decomposition.

Put your answer to the above question here.

2. Decompositions¶

image showing how a matris is just a change of basis

Animiated Image from Wikipedia: https://wikipedia.org/

In numerical linear algebra, we factorize matrices to facilitate efficient and/or accurate computations (they are also helpful in proofs). There are many possible matrix decompositions. Some, e.g., the eigendecomposition, require the matrix to be square, while others, e.g., the $QR$ factorization, exist for arbitrary matrices. Among all possible decompositions (also called factorizations), some common examples include:

QR Factorization from Gram-Schmidt orthogonization:
- $A = QR$
- $Q$ has orthonormal columns and $R$ is a upper-triangular matrix
- If there are zero rows in $R$, we can reduce the number of columns in $Q$
- Exists for arbitrary matrices
LU / LDU Decomposition from Gauss Elimination:
- $A = LU$ or $A = LDU$
- $L$ is lower-triangular, $U$ is upper-triangular, and $D$ is diagonal
- Exists for all square matrices
- Is related to Gaussian Elimination
Cholesky Decomposition:
- $A = R^TR\quad (= LDL^T)$
- $R$ is upper-triangular
- Factorization of $A$ into $R^TR$ requires $A$ be symmetric and positive-definite. The latter simply requires $x^{T}Ax > 0$ for every $x \in \mathbb{R}^n$. Note that $x^{T}Ax$ is always a scalar value (e.g., note that $x^TA = y^T$ for some vector $y\in\mathbb{R}^n$, and $y^Tx$ is the dot product between $x$ and $y$ and, hence, a real scalar).
Schur Decomposition:
- $A = UTU^{T}$
- $U$ is orthogonal and $T$ is upper-triangular
- Exists for every square matrix and says every such matrix, $A$, is unitarily equivalent to an upper-triangular matrix, $T$ (i.e., there exists an orthonomal basis with respect to which $A$ is upper-triangular)
- Eigenvalues on diagonal of $T$
Singular Value Decomposition:
- $A = U\Sigma V^{T}$
- $U$ is orthogonal, $V$ is orthogonal, and $\Sigma$ is diagonal
- Exists for arbitrary matrices
Eigenvalue Decomposition:
- $A = X\Lambda X^{-1}$
- $X$ is invertible and $\Lambda$ is diagonal
- Exists for square matrices with linearly independent columns (e.g., full rank)
- Also called the eigendecomposition

✅ **QUESTION:** What decompositions have we covered in the class so far and how did we use them?

Your answer goes here

3. Assignment wrap-up¶

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✅ **Assignment-Specific QUESTION:** What other decompositions have we covered in the class so far?