Summary of Matrix Decompositions

Here is a comparison of the key matrix decompositions related to eigenvalues and singular values.

Decomposition	Requirements for Existence	Components & Properties	Interpretation & Applications	Relation to Others
Schur Decomposition \(A = Q T Q^H\)	Exists for any square complex matrix \(A\).	Q: Unitary (\(Q^H Q = I\)). T: Upper triangular. Diagonal of T contains the eigenvalues of A.	Interpretation: Any linear transformation is unitarily similar to a triangular one. Applications: Numerically stable computation of all eigenvalues.	This is the general form from which eigendecomposition is a special case.
Real Schur Decomposition \(A = Q S Q^T\)	Exists for any square real matrix \(A\).	Q: Real orthogonal (\(Q^T Q = I\)). S: Real quasi-triangular (block upper triangular). 1x1 diagonal blocks are real eigenvalues; 2x2 blocks correspond to complex conjugate eigenvalues.	Interpretation: Represents any real transformation in a basis that makes it “as triangular as possible” using only real numbers. Applications: Stable eigenvalue computation in real arithmetic.	The real-valued equivalent of the complex Schur decomposition.
Eigendecomposition \(A = X \Lambda X^{-1}\)	Exists only for diagonalizable matrices (those with \(n\) linearly independent eigenvectors).	X: Invertible matrix whose columns are the eigenvectors. \(\Lambda\): Diagonal matrix of eigenvalues. \(X\) is not guaranteed to be unitary.	Interpretation: The transformation is a pure scaling along the eigenvector axes. Associated with “Time” and the evolution of dynamical systems (\(A^k\)). Applications: Solving linear differential equations, analyzing stability, computing matrix powers.	A special case of the Schur decomposition where the triangular matrix \(T\) is diagonal.
Unitary Diagonalization \(A = Q \Lambda Q^H\)	Exists only for normal matrices (\(A A^H = A^H A\)).	Q: Unitary matrix whose columns are an orthonormal basis of eigenvectors. \(\Lambda\): Diagonal matrix of eigenvalues.	Interpretation: The transformation is a pure scaling along a set of orthogonal axes. Applications: Quantum mechanics, signal processing.	The ideal form of both the Schur and eigendecomposition. Occurs when the matrix is normal.
Singular Value Decomposition (SVD) \(A = U \Sigma V^T\)	Exists for any \(m \times n\) matrix \(A\), real or complex.	U: Unitary/Orthogonal matrix (\(m \times m\)) of left singular vectors. V: Unitary/Orthogonal matrix (\(n \times n\)) of right singular vectors. \(\Sigma\): Rectangular diagonal matrix of non-negative singular values.	Interpretation: Decomposes any transformation into rotation, scaling, and another rotation. Associated with “Space” and the geometry of the transformation. Applications: PCA, low-rank approximation, computing pseudoinverses, determining rank and condition number.	Generalizes eigendecomposition to any matrix. The singular values of A are the square roots of the eigenvalues of \(A^T A\) and \(A A^T\).