Math Problem Statement
Solution
To find the eigenvalues of matrix , we need to solve the characteristic polynomial, which is obtained by setting . Here is the given matrix:
Steps to Find Eigenvalues:
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Construct , where is the eigenvalue and is the identity matrix:
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Compute the determinant of and set it equal to zero to find the characteristic polynomial:
This process generally results in a cubic polynomial equation in terms of . Solving this polynomial will give the eigenvalues of .
Solution (Eigenvalues):
After calculating the determinant and solving the characteristic polynomial, we find that the eigenvalues of (approximately) are:
These eigenvalues are listed from largest to smallest as requested.
Would you like more detailed steps for the determinant calculation or the characteristic polynomial?
Related Questions:
- What are the eigenvectors corresponding to these eigenvalues?
- How does the symmetry of matrix affect the eigenvalues?
- What methods can be used to approximate eigenvalues for larger matrices?
- How do eigenvalues relate to the stability of a system in engineering applications?
- Why do we often start with the largest eigenvalue in many applications?
Tip:
For symmetric matrices, the eigenvalues are always real, which simplifies finding them since complex solutions are not required.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Matrices
Formulas
Characteristic Polynomial: det(A - λI) = 0
Theorems
Eigenvalue Theorem
Determinant Theorem
Suitable Grade Level
Undergraduate Level