Math Problem Statement

In the notation and conventions used in the book by M. P. Deisenroth et al., the matrices

U, 2. V appearing in the singular value decomposition of a matrix A are:

$U = \begin{bmatrix} \frac{1}{\sqrt{5}} & \frac{-2}{\sqrt{5}} \ \frac{2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{bmatrix}$

$S = \begin{bmatrix} 3 & 0 & 0 \ 0 & 2 & 0 \end{bmatrix}$

$V = \begin{bmatrix} \frac{\sqrt{5}}{3} & 0 & 2/3 \ \frac{2}{3\sqrt{5}} & \frac{-2}{\sqrt{5}} & \frac{-1}{3} \ \frac{-4}{3\sqrt{5}} & \frac{-1}{\sqrt{5}} & \frac{2}{3} \end{bmatrix}$

Calculate the matrix A.

Solution

The singular value decomposition (SVD) of a matrix AA is given by A=USVTA = U S V^T, where UU, SS, and VV are the matrices provided:

  • U=[15−252515]U = \begin{bmatrix} \frac{1}{\sqrt{5}} & \frac{-2}{\sqrt{5}} \\ \frac{2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{bmatrix}
  • S=[300020]S = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 2 & 0 \end{bmatrix}
  • V=[53023235−25−13−435−1523]V = \begin{bmatrix} \frac{\sqrt{5}}{3} & 0 & \frac{2}{3} \\ \frac{2}{3\sqrt{5}} & \frac{-2}{\sqrt{5}} & \frac{-1}{3} \\ \frac{-4}{3\sqrt{5}} & \frac{-1}{\sqrt{5}} & \frac{2}{3} \end{bmatrix}

First, compute the transpose of VV, denoted as VTV^T:

\frac{\sqrt{5}}{3} & \frac{2}{3\sqrt{5}} & \frac{-4}{3\sqrt{5}} \\ 0 & \frac{-2}{\sqrt{5}} & \frac{-1}{\sqrt{5}} \\ \frac{2}{3} & \frac{-1}{3} & \frac{2}{3} \end{bmatrix}$$ Next, compute the product $$S V^T$$: $$S V^T = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 2 & 0 \end{bmatrix} \begin{bmatrix} \frac{\sqrt{5}}{3} & \frac{2}{3\sqrt{5}} & \frac{-4}{3\sqrt{5}} \\ 0 & \frac{-2}{\sqrt{5}} & \frac{-1}{\sqrt{5}} \\ \frac{2}{3} & \frac{-1}{3} & \frac{2}{3} \end{bmatrix}$$ This yields: $$S V^T = \begin{bmatrix} 3 \cdot \frac{\sqrt{5}}{3} & 3 \cdot \frac{2}{3\sqrt{5}} & 3 \cdot \frac{-4}{3\sqrt{5}} \\ 2 \cdot 0 & 2 \cdot \frac{-2}{\sqrt{5}} & 2 \cdot \frac{-1}{\sqrt{5}} \end{bmatrix} = \begin{bmatrix} \sqrt{5} & \frac{2}{\sqrt{5}} & \frac{-4}{\sqrt{5}} \\ 0 & \frac{-4}{\sqrt{5}} & \frac{-2}{\sqrt{5}} \end{bmatrix}$$ Finally, compute the product $$A = U (S V^T)$$: $$A = \begin{bmatrix} \frac{1}{\sqrt{5}} & \frac{-2}{\sqrt{5}} \\ \frac{2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{bmatrix} \begin{bmatrix} \sqrt{5} & \frac{2}{\sqrt{5}} & \frac{-4}{\sqrt{5}} \\ 0 & \frac{-4}{\sqrt{5}} & \frac{-2}{\sqrt{5}} \end{bmatrix}$$ Breaking down the multiplication: $$\begin{bmatrix} \frac{1}{\sqrt{5}} & \frac{-2}{\sqrt{5}} \\ \frac{2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{bmatrix} \begin{bmatrix} \sqrt{5} & \frac{2}{\sqrt{5}} & \frac{-4}{\sqrt{5}} \\ 0 & \frac{-4}{\sqrt{5}} & \frac{-2}{\sqrt{5}} \end{bmatrix} = \begin{bmatrix} \frac{1}{\sqrt{5}} \cdot \sqrt{5} + \frac{-2}{\sqrt{5}} \cdot 0 & \frac{1}{\sqrt{5}} \cdot \frac{2}{\sqrt{5}} + \frac{-2}{\sqrt{5}} \cdot \frac{-4}{\sqrt{5}} & \frac{1}{\sqrt{5}} \cdot \frac{-4}{\sqrt{5}} + \frac{-2}{\sqrt{5}} \cdot \frac{-2}{\sqrt{5}} \\ \frac{2}{\sqrt{5}} \cdot \sqrt{5} + \frac{1}{\sqrt{5}} \cdot 0 & \frac{2}{\sqrt{5}} \cdot \frac{2}{\sqrt{5}} + \frac{1}{\sqrt{5}} \cdot \frac{-4}{\sqrt{5}} & \frac{2}{\sqrt{5}} \cdot \frac{-4}{\sqrt{5}} + \frac{1}{\sqrt{5}} \cdot \frac{-2}{\sqrt{5}} \end{bmatrix}$$ $$= \begin{bmatrix} 1 & \frac{2}{5} + \frac{8}{5} & \frac{-4}{5} + \frac{4}{5} \\ 2 & \frac{4}{5} + \frac{-4}{5} & \frac{-8}{5} + \frac{-2}{5} \end{bmatrix}$$ $$= \begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & -2 \end{bmatrix}$$ Therefore, the matrix $$A$$ is: $$A = \begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & -2 \end{bmatrix}$$ Would you like further details or have any questions? Here are some additional questions you might consider: 1. What is the significance of singular value decomposition (SVD) in linear algebra? 2. How do the orthogonal properties of $$U$$ and $$V$$ affect the SVD of a matrix? 3. Can SVD be used to approximate a matrix with lower rank? 4. How do singular values relate to the eigenvalues of a matrix? 5. What are some practical applications of SVD in data science? **Tip:** When performing matrix multiplications, double-check each element computation to ensure accuracy.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Singular Value Decomposition (SVD)

Formulas

Singular Value Decomposition (SVD): A = U S V^T

Theorems

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Suitable Grade Level

Undergraduate

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