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MATH 218 Exam 2 (Eigenvalues) Questions And Answers Latest Update
- Course
- Math 218
- Institution
- Duke University
MATH 218 Exam 2 (Eigenvalues) Questions And Answers Latest Update
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2024/2025MATH 218 Exam 2 (Eigenvalues)
Questions And Answers Latest
Update
characteristic matrix (XA(t)) ANS✔✔ XA(t)=tIn-A
A scalar λ is an eigenvalue of A IFF XA(λ)=λIn-A is ANS✔✔ singular, meaning a square matrix
that does not have a full rank meaning the nullity>0
When the bottom row is zero this implies ANS✔✔ no matter how much row reducing we do there
will always be a null column
example: You are given a Matrix A and a possible eigenvalue. How you you confirm this eigenvalue?
ANS✔✔ 1. Create characteristic matrix λIn-A, then row reduce, then check if rank = number of
columns, and if not true then the eigenvalue is true.
sometime we can calculate rref(λIn-A) without row reducing. This can be done ANS✔✔ by
column relations
observation: if λ=0 then the rank of the characteristic matrix is ANS✔✔ less than the number of
column meaning it is singular
Theorem: the eigenvalues of a triangular matrix are ANS✔✔ the diagonal entries
upper triangular matrix ANS✔✔ 0's below the diagonal
lower triangular matrix ANS✔✔ 0's above diagonal
geometric multiplicity gma(λ)= ANS✔✔ the nullity(λIn-A)
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