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Eigenvalues & Eigenvectors
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Eigenvalues and eigenvectors defined by Av=λv, finding eigenvalues from det(A−λI)=0 and the characteristic polynomial, finding eigenvectors by solving the resulting simultaneous equations, and diagonalizing matrices as
Definition of eigenvectors & eigenvalues
AHL AI 1.15
The eigenvectors of a matrix A are the vector(s) v such that
Av=λv
for some constant(s) λ which we call eigenvalues.
Notice that any multiple of an eigenvector is also an eigenvector:
Av=λv⟹A(kv)=λ(kv)
Finding Eigenvalues
AHL AI 1.15
The eigenvalues λ of a matrix A satisfy
det(A−λI)=0
For example, if A=(−1−234) then
det(−1−λ−234−λ)=0⟹(−1−λ)(4−λ)+6=0
which simplifies to
λ2−3λ+2=0⇒λ=−1,−2
We call λ2−3λ+2 the characteristic polynomial of A.
Finding Eigenvectors
AHL AI 1.15
If we know an eigenvalue of a matrix A, we can find the corresponding eigenvector using its definition:
A=(acbd)(xy)=λ(xy).
When the matrix A is known, we can solve this system of simultaneous equations to find the eigenvector (xy).
Diagonalizing a matrix
AHL AI 1.15
If a matrix A has two distinct, real eigenvalues, then we can write it in the form
A=PDP−1
where P=(v1v2) is formed with the eigenvectors of A as its columns, and D=(λ100λ2) is a diagonal matrix.
The following animation shows how eigenvectors become the axes. The process of diagonalization is essentially changing the basis vectors to use the eigenvectors instead of (10) and (01).
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