Lab 19. Applications of Multilinear Algebra: Independent Study

This lab accompanies Chapter 19: Applications of Multilinear Algebra.

The goal is to make tensor products, Kronecker products, wedge products, and simple tensor decompositions computational and visual.

This is an independent-study lab. Each main question includes a worked solution and a similar practice question.

Python practice notebook

You may use the Jupyter notebook version for longer Python practice:

Interactive lab

Study guide and worked questions

Question 1. Outer products and simple tensors

Let

\[ u=\begin{bmatrix}1\\2\\3\end{bmatrix}, \qquad v=\begin{bmatrix}4\\5\end{bmatrix}. \]

Compute the matrix corresponding to \(u\otimes v\).

Solution

Under the identification \(\mathbb R^3\otimes\mathbb R^2\cong\mathbb R^{3\times2}\),

\[ u\otimes v\leftrightarrow uv^T. \]

Thus

\[ uv^T= \begin{bmatrix}1\\2\\3\end{bmatrix} \begin{bmatrix}4&5\end{bmatrix} = \begin{bmatrix} 4&5\\ 8&10\\ 12&15 \end{bmatrix}. \]

This is a rank-one matrix because every column is a scalar multiple of \(u\).

Similar practice

Compute \(a\otimes b\) for

\[ a=\begin{bmatrix}2\\-1\end{bmatrix}, \qquad b=\begin{bmatrix}3\\0\\4\end{bmatrix}. \]

Answer

\[ ab^T= \begin{bmatrix} 6&0&8\\ -3&0&-4 \end{bmatrix}. \]

Question 2. Kronecker product

Let

\[ A=\begin{bmatrix}1&2\\3&4\end{bmatrix}, \qquad B=\begin{bmatrix}0&5\\6&7\end{bmatrix}. \]

Compute \(A\otimes B\).

Solution

Use the block definition:

\[ A\otimes B= \begin{bmatrix} 1B&2B\\ 3B&4B \end{bmatrix}. \]

Therefore

\[ A\otimes B= \begin{bmatrix} 0&5&0&10\\ 6&7&12&14\\ 0&15&0&20\\ 18&21&24&28 \end{bmatrix}. \]

Similar practice

Compute

\[ \begin{bmatrix}1&0\\0&-1\end{bmatrix} \otimes \begin{bmatrix}2&1\\1&2\end{bmatrix}. \]

Answer

\[ \begin{bmatrix} 2&1&0&0\\ 1&2&0&0\\ 0&0&-2&-1\\ 0&0&-1&-2 \end{bmatrix}. \]

Question 3. Verify the vectorization identity

Let

\[ A=\begin{bmatrix}1&2\\0&1\end{bmatrix}, \quad X=\begin{bmatrix}1&3\\2&4\end{bmatrix}, \quad B=\begin{bmatrix}2&0\\1&1\end{bmatrix}. \]

Verify

\[ \operatorname{vec}(AXB)=(B^T\otimes A)\operatorname{vec}(X). \]

Solution

First compute both sides using column-stacking convention.

\[ AX= \begin{bmatrix}1&2\\0&1\end{bmatrix} \begin{bmatrix}1&3\\2&4\end{bmatrix} = \begin{bmatrix}5&11\\2&4\end{bmatrix}. \]

Then

\[ AXB= \begin{bmatrix}5&11\\2&4\end{bmatrix} \begin{bmatrix}2&0\\1&1\end{bmatrix} = \begin{bmatrix}21&11\\8&4\end{bmatrix}. \]

So

\[ \operatorname{vec}(AXB)= \begin{bmatrix}21\\8\\11\\4\end{bmatrix}. \]

On the other hand,

\[ B^T\otimes A =\begin{bmatrix}2&1\\0&1\end{bmatrix}\otimes A =\begin{bmatrix} 2A&A\\ 0A&A \end{bmatrix}. \]

Multiplying by

\[ \operatorname{vec}(X)=\begin{bmatrix}1\\2\\3\\4\end{bmatrix} \]

gives the same vector.

Similar practice

Choose your own \(2\times2\) matrices \(A,X,B\) and verify the identity using Python.

Answer

A correct Python check should return True for np.allclose(left, right).

Question 4. Wedge product and signed area

Let

\[ u=\begin{bmatrix}2\\1\end{bmatrix}, \qquad v=\begin{bmatrix}1\\3\end{bmatrix}. \]

Compute \(u\wedge v\) and interpret the result.

Solution

\[ u\wedge v=(2\cdot3-1\cdot1)(e_1\wedge e_2)=5(e_1\wedge e_2). \]

The signed area of the parallelogram spanned by \(u\) and \(v\) is \(5\).

Similar practice

Compute the signed area for

\[ a=\begin{bmatrix}1\\4\end{bmatrix}, \qquad b=\begin{bmatrix}3\\2\end{bmatrix}. \]

Answer

\[ \det\begin{bmatrix}1&3\\4&2\end{bmatrix}=2-12=-10. \]

The signed area is \(-10\), and the ordinary area is \(10\).

Question 5. Entanglement as matrix rank

Consider the two-qubit state

\[ \psi=\frac{1}{\sqrt2}(|00\rangle+|11\rangle). \]

Decide whether it is separable.

Solution

The coefficient matrix is

\[ M=\frac{1}{\sqrt2}\begin{bmatrix}1&0\\0&1\end{bmatrix}. \]

This matrix has rank \(2\). A separable two-qubit state corresponds to a rank-one coefficient matrix. Therefore this Bell state is entangled.

Similar practice

Decide whether

\[ \phi=\frac12(|00\rangle+|01\rangle+|10\rangle+|11\rangle) \]

is separable.

Answer

The coefficient matrix is

\[ \frac12\begin{bmatrix}1&1\\1&1\end{bmatrix}, \]

which has rank \(1\). Therefore \(\phi\) is separable.

Python starter code

Code 
import numpy as np

# Outer product
u = np.array([1, 2, 3])
v = np.array([4, 5])
print(np.outer(u, v))

# Kronecker product
A = np.array([[1, 2], [3, 4]])
B = np.array([[0, 5], [6, 7]])
print(np.kron(A, B))

# Vectorization identity
A = np.array([[1, 2], [0, 1]])
X = np.array([[1, 3], [2, 4]])
B = np.array([[2, 0], [1, 1]])
left = (A @ X @ B).reshape(-1, order="F")
right = np.kron(B.T, A) @ X.reshape(-1, order="F")
print(np.allclose(left, right))
[[ 4  5]
 [ 8 10]
 [12 15]]
[[ 0  5  0 10]
 [ 6  7 12 14]
 [ 0 15  0 20]
 [18 21 24 28]]
True