De64257nition 2 Computation and Properties 3 Chains brPage 3br Generalized Eigenvectors Math 240 De64257nition Computation and Properties Chains Motivation Defective matrices cannot be diagonalized because they do not possess enough eigenvectors to ID: 24036 Download Pdf

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De64257nition 2 Computation and Properties 3 Chains brPage 3br Generalized Eigenvectors Math 240 De64257nition Computation and Properties Chains Motivation Defective matrices cannot be diagonalized because they do not possess enough eigenvectors to

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Generalized Eigenvectors Math 240 Deﬁnition Computation and Properties Chains Generalized Eigenvectors Math 240 — Calculus III Summer 2013, Session II Wednesday, July 31, 2013

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Generalized Eigenvectors Math 240 Deﬁnition Computation and Properties Chains Agenda 1. Deﬁnition 2. Computation and Properties 3. Chains

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Generalized Eigenvectors Math 240 Deﬁnition Computation and Properties Chains Motivation Defective matrices cannot be diagonalized because they do not possess enough eigenvectors to make a basis. How can we

correct this defect? Example The matrix 1 1 0 1 is defective. 1. Only eigenvalue is = 1 2. 0 1 0 0 3. Single eigenvector = (1 0) 4. We could use = (0 1) to complete a basis. 5. Notice that and Maybe we just didn’t multiply by λI enough times.

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Generalized Eigenvectors Math 240 Deﬁnition Computation and Properties Chains Deﬁnition Deﬁnition If is an matrix, a generalized eigenvector of corresponding to the eigenvalue is a nonzero vector satisfying λI for some positive integer Equivalently, it is a nonzero element of the nullspace of λI Example

Eigenvectors are generalized eigenvectors with = 1 In the previous example we saw that = (1 0) and = (0 1) are generalized eigenvectors for 1 1 0 1 and = 1

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Generalized Eigenvectors Math 240 Deﬁnition Computation and Properties Chains Computing generalized eigenvectors Example Determine generalized eigenvectors for the matrix 1 1 0 0 1 2 0 0 3 1. Characteristic polynomial is (3 )(1 2. Eigenvalues are = 1 3. Eigenvectors are = 3 : = (1 2) = 1 : = (1 0) 4. Final generalized eigenvector will a vector such that but Pick = (0 0) Note that

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Generalized

Eigenvectors Math 240 Deﬁnition Computation and Properties Chains Facts about generalized eigenvectors How many powers of λI do we need to compute in order to ﬁnd all of the generalized eigenvectors for Fact If is an matrix and is an eigenvalue with algebraic multiplicity , then the set of generalized eigenvectors for consists of the nonzero elements of nullspace λI In other words, we need to take at most powers of λI to ﬁnd all of the generalized eigenvectors for

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Generalized Eigenvectors Math 240 Deﬁnition Computation and Properties

Chains Computing generalized eigenvectors Example Determine generalized eigenvectors for the matrix 1 2 0 1 1 2 1 1 1. Single eigenvalue of = 1 2. Single eigenvector = ( 1) 3. Look at 2 0 4 0 0 0 1 0 to ﬁnd generalized eigenvector = (0 0) 4. Finally, , so we get = (1 0)

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Generalized Eigenvectors Math 240 Deﬁnition Computation and Properties Chains Facts about generalized eigenvectors The aim of generalized eigenvectors was to enlarge a set of linearly independent eigenvectors to make a basis. Are there always enough generalized eigenvectors to do so? Fact If is an

eigenvalue of with algebraic multiplicity , then nullity λI k. In other words, there are linearly independent generalized eigenvectors for Corollary If is an matrix, then there is a basis for consisting of generalized eigenvectors of

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Generalized Eigenvectors Math 240 Deﬁnition Computation and Properties Chains Computing generalized eigenvectors Example Determine generalized eigenvectors for the matrix 1 2 0 1 1 2 1 1 1. From last time, we have eigenvalue = 1 and eigenvector = ( 1) 2. Solve to get = (0 0) 3. Solve to get = ( 0)

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Generalized

Eigenvectors Math 240 Deﬁnition Computation and Properties Chains Chains of generalized eigenvectors Let be an matrix and a generalized eigenvector of corresponding to the eigenvalue . This means that λI for a positive integer If q , then λI λI That is, λI is also a generalized eigenvector corresponding to for = 0 ,...,p Deﬁnition If is the smallest positive integer such that λI then the sequence λI λI , ..., λI is called a chain or cycle of generalized eigenvectors. The integer is called the length of the cycle.

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Generalized Eigenvectors Math 240 Deﬁnition Computation and Properties Chains Chains of generalized eigenvectors Example In the previous example, λI 0 2 0 1 0 2 1 0 and we found the chain λI λI Fact The generalized eigenvectors in a chain are linearly independent.

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Generalized Eigenvectors Math 240 Deﬁnition Computation and Properties Chains Jordan canonical form What’s the analogue of diagonalization for defective matrices? That is, if ,..., are the linearly independent generalized eigenvectors of , what does the matrix AS look like, where ···

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