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The characteristic equation
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The characteristic equation
Lessons
We say that a scalar λ is an eigenvalue of an n×n matrix A if and only if λ satisfies the following characteristic equation:
det(A−λI)=0
We say that det(A−λI) is a characteristic polynomial.
Useful ways to find eigenvalues
When dealing with a 2×2 matrix, use the formula det(A)=ad−bc.
When dealing with a 3×3 matrix, use the shortcut method.
When dealing with a triangular matrix, know that the determinant is just the product of the diagonal entries.
Note that:
1. An eigenvalue is a distinct root if it has a multiplicity of 1
2. An eigenvalue is a repeated root if it has a multiplicity greater than 1
We say that det(A−λI) is a characteristic polynomial.
Useful ways to find eigenvalues
When dealing with a 2×2 matrix, use the formula det(A)=ad−bc.
When dealing with a 3×3 matrix, use the shortcut method.
When dealing with a triangular matrix, know that the determinant is just the product of the diagonal entries.
Note that:
1. An eigenvalue is a distinct root if it has a multiplicity of 1
2. An eigenvalue is a repeated root if it has a multiplicity greater than 1
- IntroductionThe Characteristic Equation Overview:a)What is the Characteristic Equation?
• det(A−λI)=0
• The characteristic polynomial is det(A−λI)b)Finding the Eigenvalue
• The eigenvalues of a 2×2 matrix
• The eigenvalues of a 3×3 matrix
• The eigenvalues of a triangular matrixc)Shortcut to Determinants of Matrices
• 2×2 matrices
• 3×3 matrices
• Triangular matricesd)Eigenvalue with Multiplicity
• Distinct eigenvalues
• Repeated eigenvalues - 1.Finding the Characteristic Polynomial
Find the characteristic polynomial of A if:
- 2.Find the characteristic polynomial of A if:
- 3.Finding the Eigenvalues of a 2×2 matrix
Find all the eigenvalues of the matrix
- 4.Finding the Eigenvalues of a 3×3 matrix
Find all the eigenvalues of the matrix
- 5.Finding the Eigenvalues of a triangular matrix
Find all the eigenvalues of the matrix
- 6.Proofs dealing with the Characteristic Equation
Show that A and AT has the same characteristic polynomials. - 7.Suppose A is an n×n triangular matrix where all the diagonal entries are c. Then the characteristic polynomial is
p(λ)=(c−λ)n