29.4. Properties of Eigenvalues

Try these first

1. 1.

   What is the trace of a matrix?

   Hint

   Count only the diagonal entries.

2. 2.

   How do you find the eigenvalues of a triangular matrix?

   Hint

   Check the diagonals of the matrix.

3. 3.

   What does the trace of a matrix represent?

   • Sum of the rows
   • Product of the eigenvalues
   • Sum of the eigenvalues

   Hint

   Think about the diagonal of the matrix.

4. 4.

   True or False: The determinant of a matrix is equal to the sum of its eigenvalues.

   • True
   • False

   Hint

   Recall the relationship between determinant and eigenvalues.

5. 5.

   Prove that the sum of the eigenvalues of a matrix equals its trace.

   Hint

   Consider how eigenvalues are derived and how they relate to matrix components.

6. 6.

   Explain why a triangular matrix simplifies the process of finding eigenvalues and apply it to find eigenvalues of a specific 2x2 triangular matrix.

   Hint

   Focus on the definitions of triangular matrices and diagonal properties.