- eigenvector or characteristic vector (𝑥) of a linear transformation (𝐴𝑥) is a nonzero vector 𝑥 that changes at most by a scalar factor (𝑖.𝑒. 𝐴𝑥=𝜆𝑥) when that linear transformation is applied to it
- eigenvalue (𝜆) is the factor by which the eigenvector is scaled
- spectrum ({𝜆1, …, 𝜆𝑟}) - the set of eigenvalues of a matrix 𝐴
Eigenvalues & Eigenvectors of a Linear Transformation Matrix 𝐴
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Definition |
Example Computations |
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Given a linear transformation square matrix 𝐴 find all of its eigenvectors and eigenvalues 1 - Solve for eigenvalues 𝜆𝑖:
2 - For each eigenvalue 𝜆𝑖 solve for its corresponding eigenvector 𝑣𝑖:
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---eigenvalues-(spectrum)/eigenvector-eigenvalue.png)
---eigenvalues-(spectrum)/eigenvalue-computation2.png)
---eigenvalues-(spectrum)/eigenvector-computation-2.png)
---eigenvalues-(spectrum)/eigenvalue-computation.png)
---eigenvalues-(spectrum)/eigenvector-computation.png)
---eigenvalues-(spectrum)/eigenvalue-computation-0.png)
---eigenvalues-(spectrum)/eigenvector-computation-0a.png)
---eigenvalues-(spectrum)/eigenvector-computation-0b.png)
Eigenvalues & Eigenvectors - Complex Numbers
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Every 𝑛×𝑛 matrix has exactly 𝑛 complex eigenvalues, counted with multiplicity.
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Subpages
- A matrix and its transpose have the same characteristic polynomial and eigenvalues
- Complex Eigenvalues - Complex Eigenvectors
- Eigenvalues are the roots of the Characteristic Polynomial
- Eigenvectors with Distinct Eigenvalues are Linearly Independent