Final Exam Problem Solving

TL;DR

This video explains how to find the third eigenvalue and third pivot of a given matrix, as well as the smallest number that would make the matrix positive semidefinite.

Transcript

PROFESSOR: Today, we're going to be solving a problem from a final exam. And here it is. It's about a matrix A, [1, 0, 1; 0, 1, 1; 1, 1, 0]. And we know that this matrix has two eigenvalues, 1 and 2. And we also know that if we do elimination, the first two pivots will be 1 and 1. And here are two questions about this matrix. The first one is find... Read More

Key Insights

• 🍹 The sum of all the eigenvalues of a matrix is equal to the trace of the matrix.
• 🫤 The determinant of an upper triangular matrix is the product of its diagonal entries.
• 🏆 The determinant test can be used to determine if a matrix is positive semidefinite.
• 🪚 Markov matrices have a unique steady state eigenvector even when there are non-zero entries.

Questions & Answers

Q: How can we find the third eigenvalue of a matrix?

To find the third eigenvalue, we can use the sum of all the eigenvalues, which is equal to the trace of the matrix. By subtracting the sum of the known eigenvalues from the trace, we can determine the third eigenvalue.

Q: How can we find the third pivot of a matrix without elimination?

The trick is to use the fact that the determinant of an upper triangular matrix is equal to the product of its diagonal entries. By equating this determinant to the determinant of the original matrix, we can solve for the third pivot.

Q: How can we determine the smallest value that would make a matrix positive semidefinite?

The matrix can be positive semidefinite when its determinant is non-negative. By using the determinant test, we can calculate the determinant of the matrix and set it greater than or equal to zero to find the smallest value that would satisfy this condition.

Q: What is the limit behavior of the iteration process in a Markov matrix?

In a Markov matrix, the limit behavior of the iteration process can be found by finding the eigenvector corresponding to the eigenvalue of 1. This eigenvector represents the unique steady state of the matrix.

Summary & Key Takeaways

• The video discusses how to find the third eigenvalue of a matrix by using the trace of the matrix and the sum of all the eigenvalues.

• It explains a trick to find the third pivot of a matrix without performing elimination by using the determinant and the product of the eigenvalues.

• The video also covers how to determine the smallest entry that would make the matrix positive semidefinite by using the determinant test.