1. Eigenvectors and Eigenvalues

2. Copyright Gene A Taglairini, PhD
Introduction
Eigenvalues and eigenvectors are also known as characteristic values and characteristic vectors
Determine the stability of linear systems
Describe characteristics of statistical distributions
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Copyright Gene A Taglairini, PhD

3. Copyright Gene A Taglairini, PhD
Some Definitions
Suppose:
An x n is a matrix,
en x 1  0 is a vector, and
l is a scalar such that Ae = le
Then
e is an eigenvector (characteristic vector) of A
l is an eigenvalue (characteristic value) of A
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Copyright Gene A Taglairini, PhD

4. Characteristic Polynomial
Ae = lIe, where I is the n x n identity matrix
(lI – A)e = 0 is a homogeneous system
|(lI – A)| is the characteristic polynomial of A
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Copyright Gene A Taglairini, PhD

5. Copyright Gene A Taglairini, PhD
Example
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6. Copyright Gene A Taglairini, PhD
Example 1 (continued)
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7. Copyright Gene A Taglairini, PhD
Example 1 (continued)
The eigenvalues of A are l1 = 1 and l2 = 2
To find eigenvectors, we need nonzero solutions to (lI – A) e = 0
Consider
(2I – A) e = 0
(1I – A) e = 0
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Copyright Gene A Taglairini, PhD

8. Example 1 (continued) Eigenvector for l=2
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9. Example 1 (continued) Eigenvector for l=1
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10. Diagonal Matrix Similar to A
Find the eigenvalues and corresponding eigenvectors of A
Form the matrix P whose columns are the eigenvectors of A
Form the diagonal matrix B whose diagonal entries are the eigenvalues of A in the order in which the eigenvectors appear in P
A = P B P-1
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Copyright Gene A Taglairini, PhD

11. Example 1 (continued) Constructing a Matrix Similar to A
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Copyright Gene A Taglairini, PhD

12. Convergence of an Iterative Linear System
Since A = P B P-1
Hence Ak = P Bk P-1 (Why?)
If A is a residual arising from the application of a linear transformation, and if the eigenvalues of A are all in absolute value less than 1, then residual Ak → 0, as k → ∞
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13. Example 2: Practice Finding the Eigenvalues of a Covariance Matrix
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14. Example 2: Practice Finding the Eigenvalues of a Covariance Matrix
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15. Example 2: Practice Finding the Eigenvalues of a Covariance Matrix
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16. Example 2: Finding the Eigenvectors of a Matrix—With l1 = 8, Find e1
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17. Example 2: Finding the Eigenvectors of a Matrix—With l1 = 8, Find e1
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18. Example 2: Finding the Eigenvectors of a Matrix—With l2 = 2, Find e2
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19. Example 2: Finding the Eigenvectors of a Matrix—With l2 = 2, Find e2
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20. Example 2: Finding the Eigenvectors of a Matrix—With l3 = 1, Find e3
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21. Example 2: Finding the Eigenvectors of a Matrix—With l3 = 1, Find e3
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22. Example 2: Determining the Matrices B, P, and P-1
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23. Example 2: Determining P-1
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24. Copyright Gene A Taglairini, PhD
Example 2: P B P-1 =?
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25. Copyright Gene A Taglairini, PhD
Example 2: P B P-1 = A
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26. Copyright Gene A Taglairini, PhD
The Trace of A
The trace of a matrix A, written trace(A), is the sum of its diagonal entries, i.e.,
Let l1, l2, …, ln be the eigenvalues of A, then l1+ l2+ …+ ln = trace(A)
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Copyright Gene A Taglairini, PhD

27. The Relationship of the Determinant of A to the Eigenvalues of A
Let l1, l2, …, ln be the eigenvalues of A
Then l1* l2* …* ln = |A|
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Copyright Gene A Taglairini, PhD