SLIDE 1
Algebraic Multiplicity
Defn. The algebraic multiplicity of eigen- value λ is its multiplicity as a root of the charac- teristic polynomial. Fact. The dimension of the eigenspace of λ is at most its algebraic multiplicity.
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Properties of Eigenvalues and Eigenvectors Algebraic Multiplicity - - PowerPoint PPT Presentation
Properties of Eigenvalues and Eigenvectors Algebraic Multiplicity - - PowerPoint PPT Presentation
Properties of Eigenvalues and Eigenvectors Algebraic Multiplicity Defn. The algebraic multiplicity of eigen- value is its multiplicity as a root of the charac- teristic polynomial. Fact. The dimension of the eigenspace of is at most its
SLIDE 2
SLIDE 3
Eigenvectors and Linear Independence
- Fact. Eigenvectors for distinct eigenvalues are
linearly independent.
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SLIDE 4
Eigenvalues of Matrix Powers
Fact. If matrix A has eigenvalues λi, then the power Ak has eigenvalues λk
i .
Moreover, the eigenvectors are the same.
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SLIDE 5
Trace and Determinant
The trace of a matrix is defined as the sum of the diagonal entries. Fact. For any matrix A, (a) the determinant of A equals the product of its eigenvalues. (b) the trace of A equals the sum of its eigenval- ues.
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SLIDE 6
Complex Conjugates
Recall that i denotes the square-root of −1. Defn. If λ = a + bi, then its (complex) conju- gate is a − bi.
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SLIDE 7
Complex Eigenvalues
Fact. If λ is a complex eigenvalue of A, then so is its conjugate.
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SLIDE 8
An Example
Consider the matrix
- a −b
b a
- .
The characteristic polynomial is (a − λ)2 + b2; eigenvalues are λ = a ± bi. As a matrix transform, this represents scaling by √ a2 + b2 and rotation through arctan b/a.
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SLIDE 9
Symmetric Matrices
Fact. A real symmetric matrix has only real eigenvalues.
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SLIDE 10
Summary
The algebraic multiplicity of eigenvalue is its mul- tiplicity as root of the characteristic polynomial; the eigenspace has dimension at most its alge- braic multiplicity. Eigenvectors for distinct eigenvalues are linearly independent. If A has eigenvalue λi, then Ak has eigenvalue λk
i with same eigenvector.
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SLIDE 11
Summary (cont)
The product of the eigenvalues is the determi- nant; the sum of the eigenvalues is the trace, which is the sum of the diagonal entries. If λ is complex eigenvalue of real matrix, then so is its conjugate. A real symmetric matrix has real eigenvalues.
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