The eigenvalue of the inverse of a matrix is simply the reciprocal of the eigenvalue of the original matrix.
Here's a breakdown:
- Eigenvalues are special numbers associated with a matrix that represent scaling factors for the corresponding eigenvectors.
- Eigenvectors are non-zero vectors that remain in the same direction when multiplied by a matrix.
- Inverse of a matrix is a matrix that, when multiplied with the original matrix, results in the identity matrix.
Let's illustrate this with an example:
Suppose a matrix A has an eigenvalue λ and an eigenvector v. This means:
Av = λv
Now, consider the inverse of matrix A, denoted as A⁻¹. Multiplying both sides of the equation by A⁻¹:
A⁻¹(Av) = A⁻¹(λv)
Since A⁻¹ A equals the identity matrix (I), we get:
Iv = λ(A⁻¹v)
Simplifying further:
v = λ(A⁻¹v)
Therefore, A⁻¹ has an eigenvalue of 1/λ and the same eigenvector v.
In summary:
- The eigenvalues of the inverse of a matrix are the reciprocals of the eigenvalues of the original matrix.
- The eigenvectors remain the same for both the original matrix and its inverse.
