What’s the definition of normal linear transformation?
A linear transformation ( A ) on a vector space is called normal if it commutes with its adjoint (or conjugate transpose). Mathematically, a linear transformation ( A ) is normal if:
[ AA^* = A^*A, ]
where ( A^* ) denotes the adjoint (or conjugate transpose) of ( A ).
why it’s called normal? because the eigenvectors belongs to these eigenvalues are orthogonal/normal.
Examples are
- Hermitian Matrices:
- A matrix ( A ) is Hermitian (or self-adjoint) if ( A = A^* ). All Hermitian matrices are normal because ( AA^* = A^*A = A^2 ).
- Unitary Matrices:
- A matrix ( A ) is unitary if ( A^A = AA^ = I ), where ( I ) is the identity matrix. Unitary matrices are also normal because they commute with their adjoints.
- Diagonal Matrices:
- A diagonal matrix is normal because the adjoint of a diagonal matrix is the matrix itself with the entries conjugated, and diagonal matrices commute with their conjugates.
- Commuting Matrices Example:
- Consider ( A ) and ( B ) as commuting matrices, then ( A ) is normal if ( A = B ) and ( B = A ).
There are two practical implications/applications:
- Spectral Theorem: Normal matrices have a particularly nice spectrum. The spectral theorem states that a normal matrix can be diagonalized by a unitary matrix. This means there exists a unitary matrix ( U ) such that:
[ A = U \Lambda U^*, ]
where ( \Lambda ) is a diagonal matrix containing the eigenvalues of ( A ). - Eigenvalues: The eigenvalues of a normal matrix are well-behaved, and the matrix can be fully described by its eigenvalues and eigenvectors.
What is isometric?
An isometric transformation is a mapping that preserves the norm of vectors, implying that distances between vectors remain unchanged under the transformation. In finite-dimensional spaces, isometric transformations that are surjective are also inner product-preserving and are known as unitary (in complex spaces) or orthogonal (in real spaces) operators.
Third point, we say a linear transformation A can be dragonized by a unitary similarity transformation P iff A is normal D = P^-1 A P.
Naixian Zhang 