Question

If \(A\) has singular values \(\sigma_{1}, \ldots, \sigma_{r},\) what are the singular values of: a. \(A^{T}\) b. \(t A\) where \(t>0\) is real c. \(A^{-1}\) assuming \(A\) is invertible.

Short answer

a. Same as \(A\); b. Scaled by \(t\); c. Reciprocals.

Step-by-step solution

Understanding Singular Values

Singular values of a matrix \(A\) are the non-negative square roots of the eigenvalues of \(A^TA\) or \(AA^T\). If \(A\) is an \(m \times n\) matrix, and its singular values are \(\sigma_1, \sigma_2, \ldots, \sigma_r\) (sorted in decreasing order), they are the diagonal entries of \(\Sigma\) in the Singular Value Decomposition (SVD) \(A = U \Sigma V^T\).

Singular Values of Transpose (a)

The singular values of \(A^T\) are the same as those of \(A\). This is because the singular values are derived from the eigenvalues of \(A^TA\), and \(A^TA\) has the same eigenvalues as \(AA^T\). Thus, the singular values of \(A^T\) are \(\sigma_1, \sigma_2, \ldots, \sigma_r\).

Singular Values of Scaled Matrix (b)

For a real number \(t > 0\), the singular values of \(tA\) are obtained by scaling the singular values of \(A\) by \(t\). Hence, the singular values of \(tA\) are \(t\sigma_1, t\sigma_2, \ldots, t\sigma_r\).

Singular Values of Inverse (c)

Assuming \(A\) is invertible, the singular values of \(A^{-1}\) are the reciprocals of the singular values of \(A\). Thus, they are \(1/\sigma_1, 1/\sigma_2, \ldots, 1/\sigma_r\). This is because the singular values of \(A^{-1}\) correspond to the eigenvalues of \((A^{-1})^TA^{-1}\), which are the reciprocals of the eigenvalues of \(A^TA\).

Key concepts

Matrix Transpose

A matrix transpose plays a crucial role in many mathematical operations, including singular value decomposition (SVD). When we transpose a matrix, we swap its rows and columns. Mathematically, the transpose of a matrix \(A\), denoted as \(A^T\), has its entry at position \((i, j)\) switched with the entry at \((j, i)\). This simple operation maintains the original number of columns and rows, effectively flipping the matrix over its diagonal.

Understanding the matrix transpose is essential when working with singular values. Interestingly, for any matrix \(A\), its transpose \(A^T\) shares the same singular values as \(A\) itself. This property stems from the fact that these values are derived from the eigenvalues of \(A^TA\), which remain unaffected by transposing \(A\).

• Transpose operation swaps rows and columns.
• Transposing does not alter singular values.
• Essential in matrix operations like SVD.

Keep in mind that while the singular values remain unchanged, the matrices \(U\) and \(V\) in the decomposition \(A = U \Sigma V^T\) adjust to reflect the transpose. This concept is foundational in applications like machine learning and data compression.

Matrix Scaling

Matrix scaling involves multiplying each element of a matrix by a constant factor. This operation can be quite straightforward yet powerful, especially when dealing with singular value decomposition. Suppose we have a constant \(t > 0\). To perform matrix scaling, we simply multiply each entry of matrix \(A\) by \(t\), resulting in a new matrix \(tA\).

What's fascinating about matrix scaling is how it impacts singular values. When a matrix \(A\) is scaled by a factor \(t\), the singular values of the scaled matrix \(tA\) are \(t\) times the original singular values of \(A\). In other words, if the original matrix \(A\) has singular values \(\sigma_1, \sigma_2, \ldots, \sigma_r\), the scaled matrix \(tA\) will have singular values \(t\sigma_1, t\sigma_2, \ldots, t\sigma_r\).

• Each element is multiplied by a constant \(t\).
• Singular values are scaled by \(t\).
• Simplifies understanding impact of scaling in linear transformations.

This property is particularly useful in adjusting the scale of transformations and simplifies certain operations in engineering and computer science fields.

Matrix Inverse

The concept of a matrix inverse is pivotal in solving systems of linear equations. The inverse of a matrix \(A\), denoted \(A^{-1}\), is a matrix that, when multiplied with \(A\), yields the identity matrix. Not all matrices have inverses; invertibility is a property of square matrices that have full rank.

For singular value decomposition, the relationship between a matrix and its inverse is highly insightful. If \(A\) is invertible and has singular values \(\sigma_1, \sigma_2, \ldots, \sigma_r\), the inverse \(A^{-1}\) will have singular values that are the reciprocals of those of \(A\): \(1/\sigma_1, 1/\sigma_2, \ldots, 1/\sigma_r\). This relationship emerges because inverting the matrix essentially reverses the transformations associated with its singular values.

• Involving reciprocal of singular values in inversion.
• Inverses exist only for square, full-rank matrices.
• Key role in solving linear equations and optimizing problems.

Understanding these concepts is crucial for advanced computational algorithms and is integral to areas such as numerical analysis and regression analysis.