📚 Topic Summary
Singular Value Decomposition (SVD) is a powerful matrix factorization technique used in various applications like data compression, dimensionality reduction, and recommendation systems. It decomposes a matrix into three matrices: $U$, $\Sigma$, and $V^T$, where $U$ and $V$ are orthogonal matrices and $\Sigma$ is a diagonal matrix containing the singular values of the original matrix. Understanding SVD is crucial for anyone working with linear algebra and data science. This worksheet provides practice problems to help you grasp the concepts effectively.
🧠 Part A: Vocabulary
Match the following terms with their correct definitions:
- Term: Singular Value
- Term: Orthogonal Matrix
- Term: Eigenvector
- Term: Rank of a Matrix
- Term: Transpose
- Definition: A square matrix whose transpose is also its inverse.
- Definition: A measure of the non-degeneracy of the system of equations and linear transformations represented by the matrix.
- Definition: A vector that, when multiplied by a matrix, results in a scaled version of itself.
- Definition: The square root of the eigenvalues of $A^TA$.
- Definition: An operation which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix.
✍️ Part B: Fill in the Blanks
Complete the following paragraph with the correct terms:
Singular Value Decomposition decomposes a matrix $A$ into three matrices: $U$, $\Sigma$, and $V^T$. The matrix $U$ contains the left ___________ of $A$, while $V$ contains the right ___________ of $A$. The matrix $\Sigma$ is a ___________ matrix containing the ___________ values of $A$. The singular values are the square roots of the eigenvalues of __________.
🤔 Part C: Critical Thinking
Explain in your own words how SVD can be used for image compression, and what are the advantages and disadvantages of this method.