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Eigenvalues & Eigenvectors
Find eigenvalues & eigenvectors, and using them to diagonalize and find powers of matrices.
Find eigenvalues & eigenvectors, and using them to diagonalize and find powers of matrices.
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The eigenvectors of a matrix A are the vector(s) v such that
for some constant(s) λ which we call eigenvalues.
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.