The characteristic equation - Eigenvalue and Eigenvectors

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The characteristic equation


We say that a scalar λ\lambda is an eigenvalue of an n×nn \times n matrix AA if and only if λ\lambda satisfies the following characteristic equation:
det(AλI)=0\det(A-\lambda I)=0

We say that det(AλI)\det(A-\lambda I) is a characteristic polynomial.

Useful ways to find eigenvalues
When dealing with a 2×22 \times 2 matrix, use the formula det(A)=adbc\det (A) = ad-bc.

When dealing with a 3×33 \times 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
  • Intro Lesson
    The Characteristic Equation Overview:
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The characteristic equation

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