15. Matrices A(t) Depending on t, Derivative = dA/dt

TL;DR

Adding a rank 1 positive semidefinite matrix to a symmetric matrix increases the eigenvalues, while respecting an interlacing property.

Transcript

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Key Insights

• 💱 The derivative of the inverse matrix gives insight into how changes in the original matrix affect the inverse matrix.
• 😜 Adding a rank 1 positive semidefinite matrix to a symmetric matrix increases the eigenvalues and preserves an interlacing property.
• ❓ The interlacing property ensures that the eigenvalues of the modified matrix do not exceed the original eigenvalues significantly.

Questions & Answers

Q: How does the derivative of the inverse matrix relate to changes in the original matrix?

The derivative of the inverse matrix captures the rate of change of the original matrix. It can be computed by multiplying the inverse matrix by the derivative of the original matrix multiplied by the inverse matrix itself.

Q: What happens to eigenvalues when a rank 1 positive semidefinite matrix is added to a symmetric matrix?

The eigenvalues of the symmetric matrix increase after the addition, while respecting an interlacing property. This means that each eigenvalue of the modified matrix is greater or equal to the corresponding eigenvalue of the original matrix.

Q: Can the eigenvalues of the modified matrix exceed the eigenvalues of the original matrix by a significant margin?

In the case of adding a rank 1 positive semidefinite matrix, the eigenvalues of the modified matrix do not exceed the eigenvalues of the original matrix significantly. They increase but maintain an interlacing relationship.

Q: What happens to eigenvalues when a rank 2 positive semidefinite matrix is added to a symmetric matrix?

The eigenvalues of the modified matrix continue to increase, but the second eigenvalue of the modified matrix has the potential to surpass the first eigenvalue of the original matrix. However, subsequent eigenvalues continue to interlace with the original eigenvalues.

Summary & Key Takeaways

• The lecture focuses on understanding changes in eigenvalues and singular values when a matrix undergoes changes.

• The first part of the lecture discusses the derivative of the inverse matrix, providing a formula for infinitesimal changes.

• The second part of the lecture analyzes the changes in eigenvalues when a rank 1 positive semidefinite matrix is added to a symmetric matrix, highlighting the interlacing property.

• The lecture also poses a question about the validity of interlacing when the added matrix shares an eigenvector with the original matrix.