How to Compute Singular Values via A^TA for Rectangular Matrices

📚 Understanding Singular Values and $A^TA$

Singular Value Decomposition (SVD) is a fundamental concept in linear algebra, allowing us to decompose any matrix into a set of singular values and vectors. For a rectangular matrix $A$, the singular values are closely related to the eigenvalues of the matrix $A^TA$. Computing singular values via $A^TA$ provides a practical approach, especially when dealing with smaller matrices.

📜 Historical Context

The development of SVD can be traced back to the work of Eugenio Beltrami and Camille Jordan in the late 19th century. However, it was later refined and popularized by Gene Golub and William Kahan in the 1960s. The method of using $A^TA$ to compute singular values has been a staple in numerical linear algebra due to its simplicity and directness.

🔑 Key Principles

• ➕ Form $A^TA$: Given a rectangular matrix $A$, compute the matrix product $A^TA$. This results in a square, symmetric matrix.
• 🔢 Eigenvalue Decomposition: Find the eigenvalues $\lambda_i$ of the matrix $A^TA$. These eigenvalues are non-negative.
• √ Singular Values: The singular values $\sigma_i$ of the matrix $A$ are the square roots of the eigenvalues of $A^TA$. That is, $\sigma_i = \sqrt{\lambda_i}$.
• Eigenvectors: Determine the eigenvectors of $A^TA$. These eigenvectors form an orthonormal basis and are the right singular vectors of A.

🧮 Example Calculation

Let's consider a matrix:

$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}$

1. Compute $A^TA$:

$A^TA = \begin{bmatrix} 1 & 3 & 5 \\ 2 & 4 & 6 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix} = \begin{bmatrix} 35 & 44 \\ 44 & 56 \end{bmatrix}$

2. Find Eigenvalues:

The characteristic equation is given by $det(A^TA - \lambda I) = 0$. Solving for $\lambda$, we get the eigenvalues $\lambda_1 \approx 0.85$ and $\lambda_2 \approx 90.15$.

3. Compute Singular Values:

The singular values are $\sigma_1 = \sqrt{\lambda_1} \approx 0.92$ and $\sigma_2 = \sqrt{\lambda_2} \approx 9.49$.

💡 Real-World Applications

• 🖼️ Image Compression: SVD is used to reduce the storage space required for images by retaining only the most significant singular values.
• 📊 Data Analysis: In data science, SVD (and its variant PCA) is used for dimensionality reduction, feature extraction, and noise reduction.
• 🤖 Recommender Systems: SVD is employed in collaborative filtering techniques to predict user preferences based on historical data.

📝 Conclusion

Computing singular values via $A^TA$ is a fundamental technique in linear algebra with diverse applications across various fields. Understanding the underlying principles and steps involved allows for effective utilization of SVD in practical problems.