Spectral Decomposition

The spectral decomposition of a real symmetric matrix $\mathbf{A}$ expresses it as $\mathbf{A}=\mathbf{Q\Lambda }{\mathbf{Q}}^{\top }$, where $\mathbf{Q}$ is an orthogonal matrix of eigenvectors (so ${\mathbf{Q}}^{-1}={\mathbf{Q}}^{\top }$) and $\mathbf{\Lambda }$ is a diagonal matrix of eigenvalues -- guaranteed by the spectral theorem to exist for every real symmetric matrix, with all eigenvalues real and the eigenvectors chosen mutually orthogonal. Applied to a covariance matrix $\mathbf{\Sigma }=\mathbf{Q\Lambda }{\mathbf{Q}}^{\top }$, this is precisely the mathematics behind principal component analysis: the columns of $\mathbf{Q}$ are the principal component directions (loadings), and the eigenvalues in $\mathbf{\Lambda }$ are the variances explained along each of those directions, ordered from largest to smallest to rank the components by importance.

• Unlike a general eigendecomposition, the spectral decomposition never runs into complex eigenvalues or a missing eigenvector -- the spectral theorem guarantees both real eigenvalues and a full orthogonal eigenvector basis for every real symmetric matrix, which is exactly why symmetric matrices (covariance matrices chief among them) are so much more tractable than general matrices.
• Because $\mathbf{Q}$ is orthogonal, ${\mathbf{Q}}^{-1}={\mathbf{Q}}^{\top }$ in the spectral decomposition, no separate matrix inversion is ever needed to reconstruct $\mathbf{A}$ from its decomposition or to reverse the transformation, a computational and conceptual simplification not available for a general (non-symmetric) matrix's eigendecomposition.
• PCA is exactly the spectral decomposition of a covariance (or correlation) matrix: the eigenvectors in $\mathbf{Q}$ are the principal component loadings, mutually orthogonal directions in the original variable space, and the eigenvalues in $\mathbf{\Lambda }$, conventionally sorted largest to smallest, give the variance explained along each corresponding principal component.