A nonnegative scalar equal to the square root of an eigenvalue of A*A^* (or A^*A) for a matrix A, representing the magnitude by which A stretches or compresses unit vectors along orthogonal directions.

Principle

Singular values quantify the size of a linear operator along orthogonal input and output directions; they are invariant under pre- and post-multiplication by unitary (or orthogonal) transformations and equal the l2 norms of the action on corresponding singular vectors.

Demonstration

For A = [[3,0],[0,2]] the singular values are 3 and 2 (square roots of eigenvalues of A^TA). For a rotation matrix R, all singular values equal 1 because R preserves vector lengths. For a rank-one matrix u v^T with ||u||=||v||=1 the single nonzero singular value equals the magnitude of the matrix (the Euclidean inner product scale).

Misapplication

Treating singular values as signed or complex scaling factors (like eigenvalues) or assuming they indicate direction rather than magnitude. Using a singular value computed in one norm without adjusting when other norms or nonorthonormal bases are relevant.

Consequence

Knowing singular values yields the spectral (operator) norm, Frobenius norm (as l2 norm of the singular values), rank (number of nonzero singular values), and stability indicators for numerical algorithms and inverses.

Reversal

Interpreting the quantity as an eigenvalue of A rather than of A^*A: eigenvalues can be negative or complex and encode direction-preserving scalings, whereas singular values are nonnegative magnitudes that ignore orientation (phase).

Boundary

Defined for linear operators on inner-product spaces (finite-dimensional matrices and Hilbert–Schmidt operators); singular values require a conjugate-transpose structure and do not directly generalize to arbitrary nonlinear maps without linearization.

Semantic Tension

Singular values are often compared with eigenvalues: both measure scaling but eigenvalues include sign/phase and require an invariant direction, while singular values measure magnitude independent of invariant directions; confusion arises when one substitutes the two without noting this distinction.

Synthesis

A singular value is the nonnegative magnitude of a linear map along an orthonormal input/output direction, computed as the square root of an eigenvalue of A^*A, and it organizes norms, rank, and conditioning information into a single scalar summary.