The spectral theorem is the strongest bridge between algebra and orthogonal geometry: normal operators are exactly those that admit an orthonormal eigenbasis.
Adjoints and normality
Definition. On a finite-dimensional inner product space, the adjoint \(T^*\) is defined by \[ \langle Tv,w\rangle=\langle v,T^*w\rangle. \] An operator is self-adjoint if \(T=T^*\), unitary if \(T^*T=I\), and normal if \(T^*T=TT^*\).
Example. Real symmetric matrices, complex Hermitian matrices, unitary matrices, and skew-Hermitian matrices are normal. The matrix \(\begin{pmatrix}1&1\\0&1\end{pmatrix}\) is not normal; its failure to be normal is tied to its failure to have an orthonormal eigenbasis.
The complex spectral theorem
Theorem. If \(V\) is a finite-dimensional complex inner product space, then \(T\) is normal if and only if \(V\) has an orthonormal basis of eigenvectors of \(T\). Equivalently, a matrix \(A\) is normal if and only if \[ A=U\Lambda U^* \] for a unitary \(U\) and diagonal \(\Lambda\).
Proof. Since \(\mathbb C\) is algebraically closed, \(T\) has an eigenvector. If \(Tv=\lambda v\) and \(T\) is normal, then \(T^*v=\overline{\lambda}v\); this follows from \(\|(T- \lambda I)v\|=\|(T- \lambda I)^*v\|\). The orthogonal complement of \(v\) is invariant under both \(T\) and \(T^*\), so induction gives an orthonormal eigenbasis. The converse is immediate for diagonal matrices in an orthonormal basis.
Self-adjoint operators
Corollary. If \(T=T^*\), then all eigenvalues of \(T\) are real and eigenspaces belonging to distinct eigenvalues are orthogonal. Thus a real symmetric matrix has an orthonormal basis of real eigenvectors.
Proof. If \(Tv=\lambda v\), then \(\lambda\langle v,v\rangle=\langle Tv,v\rangle=\langle v,Tv\rangle=\overline{\lambda}\langle v,v\rangle\), so \(\lambda\) is real. If \(Tu=\lambda u\) and \(Tv=\mu v\), then \(\lambda\langle u,v\rangle=\langle Tu,v\rangle=\langle u,Tv\rangle=\mu\langle u,v\rangle\), so \(\langle u,v\rangle=0\) when \(\lambda\ne\mu\).
Quadratic forms and extrema
Proposition. If \(A=A^*\) has eigenvalues \(\lambda_1\le\cdots\le\lambda_n\), then for \(\|x\|=1\), \[ \lambda_1\le x^*Ax\le \lambda_n. \] The minimum and maximum are attained at eigenvectors for \(\lambda_1\) and \(\lambda_n\).
Proof. Write \(x=\sum_i c_iq_i\) in an orthonormal eigenbasis. Then \[ x^*Ax=\sum_i\lambda_i|c_i|^2, \qquad \sum_i|c_i|^2=1. \] This is a weighted average of the eigenvalues.
Functional calculus
Proposition. If \(A=U\Lambda U^*\) is normal and \(f\) is a polynomial or a function defined on the spectrum, then \[ f(A)=Uf(\Lambda)U^*, \] where \(f(\Lambda)\) is diagonal with entries \(f(\lambda_i)\). This defines square roots, exponentials, and spectral projections for normal matrices.
Remark. Spectral theorems are diagonalization theorems plus orthogonality. Ordinary diagonalization changes coordinates; unitary diagonalization changes coordinates without distorting lengths or angles.
Inductive proof of unitary diagonalization
Spectral theorem. A complex operator \(T\) is normal, \(T^*T=TT^*\), if and only if there is an orthonormal basis of eigenvectors.
Proof. The characteristic polynomial has a root \(\lambda\), so choose a unit eigenvector \(v\). Normality implies \[ \|(T^*-\overline\lambda I)v\|^2 =\|(T-\lambda I)v\|^2=0, \] hence \(T^*v=\overline\lambda v\). Therefore \(v^\perp\) is \(T\)-invariant: for \(x\perp v\), \[ \langle Tx,v\rangle=\langle x,T^*v\rangle=0. \] The restriction to \(v^\perp\) remains normal. Induction produces an orthonormal eigenbasis. Conversely, a unitarily diagonal matrix commutes with its adjoint.
Functional calculus and spectral projectors
If the distinct eigenvalues are \(\lambda_1,\ldots,\lambda_r\), then \[ T=\sum_{j=1}^r\lambda_jP_j,\qquad P_iP_j=\delta_{ij}P_j,\qquad \sum_jP_j=I, \] where \(P_j\) is orthogonal projection onto the \(\lambda_j\)-eigenspace. For any function \(f\) on the finite spectrum, \[ f(T)=\sum_jf(\lambda_j)P_j. \] This simultaneously defines powers, exponentials, square roots of positive operators, and spectral thresholding.