Spectral Theorem
The spectral theorem states that every normal matrix can be diagonalized by a unitary matrix, meaning that it possesses a complete orthonormal set of eigenvectors. This theorem is fundamental in linear algebra and is used to simplify operators in complex vector spaces.
Real Orthogonal Eigenvector Basis
For real matrices, having an orthogonal eigenvector basis that is also real is equivalent to the matrix being symmetric. This situation reflects the real version of the spectral theorem, whereby a real symmetric matrix can be diagonalized by an orthogonal matrix, ensuring all eigenvalues and eigenvectors are real.
Schur Decomposition
Schur Decomposition is a method that represents any square matrix as a product involving a unitary matrix and an upper triangular matrix. For normal matrices, the Schur decomposition shows that the upper triangular matrix is necessarily diagonal, hence providing a clear link between normality and the existence of an orthonormal eigenbasis.
Upper Triangular and Diagonal Matrices
An upper triangular matrix is one in which all the entries below the main diagonal are zero. A key result is that an upper triangular matrix is normal if and only if it is diagonal, because any nonzero off-diagonal entry would prevent the matrix from commuting with its conjugate transpose.
Unitary Matrix
A unitary matrix is defined by the condition that its conjugate transpose is its inverse, that is, U† U = UU† = I. This property signifies that unitary matrices preserve the inner product and norm, and it immediately implies that they are normal matrices.
Real Symmetric Matrix
A real symmetric matrix is one that is equal to its transpose. In the complex framework, such matrices are a special case of Hermitian matrices. Their symmetry immediately implies they are normal because the property of being equal to one’s own transpose ensures the commutation with the Hermitian transpose.
Hermitian Transpose
The Hermitian transpose of a matrix is obtained by taking the transpose of the matrix and then the complex conjugate of each entry. It is central to many areas of linear algebra, especially in the definitions of Hermitian, unitary, and normal matrices, since it shows how matrices interact with the inner product structure of complex vector spaces.
Normal Matrix
A normal matrix is a square matrix that commutes with its Hermitian (conjugate) transpose, meaning that A A† = A† A. This property is critical because it guarantees the matrix has a complete set of orthonormal eigenvectors, which simplifies many problems in linear algebra and quantum mechanics.
Orthogonal Matrix
An orthogonal matrix is the real analogue of a unitary matrix. It satisfies Q^T Q = QQ^T = I, meaning its transpose is its inverse. Orthogonal matrices are normal because they preserve the standard Euclidean inner product and have real eigenvalues when they are symmetric.