Eigenvalues and eigenvectors defined by Av=λv, finding eigenvalues from det(A−λI)=0
The eigenvectors of a matrix A are the vector(s) v such that
for some constant(s) λ which we call eigenvalues.
Notice that any multiple of an eigenvector is also an eigenvector:
The eigenvalues λ of a matrix A satisfy
For example, if A=(−1−234) then
which simplifies to
We call λ2−3λ+2 the characteristic polynomial of A.
If we know an eigenvalue of a matrix A, we can find the corresponding eigenvector using its definition:
When the matrix A is known, we can solve this system of simultaneous equations to find the eigenvector (xy).
If a matrix A has two distinct, real eigenvalues, then we can write it in the form
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).
Nice work completing Eigenvalues & Eigenvectors, here's a quick recap of what we covered:
Exercises checked off
Eigenvalues and eigenvectors defined by Av=λv, finding eigenvalues from det(A−λI)=0
The eigenvectors of a matrix A are the vector(s) v such that
for some constant(s) λ which we call eigenvalues.
Notice that any multiple of an eigenvector is also an eigenvector:
The eigenvalues λ of a matrix A satisfy
For example, if A=(−1−234) then
which simplifies to
We call λ2−3λ+2 the characteristic polynomial of A.
If we know an eigenvalue of a matrix A, we can find the corresponding eigenvector using its definition:
When the matrix A is known, we can solve this system of simultaneous equations to find the eigenvector (xy).
If a matrix A has two distinct, real eigenvalues, then we can write it in the form
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).
Nice work completing Eigenvalues & Eigenvectors, here's a quick recap of what we covered:
Exercises checked off