The Cayley-Hamilton Theorem Let V be a finite dimensional vector space and let P be a linear transformation on V. The Cayley-Hamilton theorem states...
Let V be a finite dimensional vector space and let P be a linear transformation on V.
The Cayley-Hamilton theorem states the following:
P is normal (P^T P = I) if and only if P is invertible and P^{-1} = P^T.
In other words:
A normal linear transformation is one that is self-adjoint (P^T P = I).
A linear transformation is invertible if and only if it is normal.
The inverse of a normal linear transformation is equal to its transpose.
Examples:
A linear transformation that projects onto a subspace is normal.
This is because the projection operator is self-adjoint and its inverse is the orthogonal projection onto the orthogonal complement of the subspace.
An orthogonal transformation is normal if and only if it is invertible.
This is because the transpose of an orthogonal transformation is its inverse.
A diagonal transformation is normal if and only if it is invertible.
This is because the transpose of a diagonal transformation is its inverse.
Applications of the Cayley-Hamilton theorem:
If you have a linear equation in a vector, you can solve for the vector by normalizing the equation (setting it equal to I) and then finding the inverse of the coefficient matrix.
For a normal linear transformation, the eigenvalues and eigenvectors are the same as for the original linear transformation.
If a linear transformation is normal, it is invertible and its inverse is equal to its transpose.
The Cayley-Hamilton theorem is a powerful tool for understanding the behavior of linear transformations and solving linear equations.