Singular value decomposition (SVD) is the piece of linear algebra that most cleanly turns abstract statements into geometry you can draw and computations you can trust. It tells you what a matrix does to the unit sphere, how far it stretches in each principal direction, and which directions are crushed nearly to zero. That single […]

The spectral theorem for real symmetric matrices is the hinge that turns linear algebra into an analytic tool. It is not just the statement that a matrix can be diagonalized. It is a complete description of how a symmetric linear map acts on space: every direction decomposes into orthogonal eigendirections, and the matrix scales each […]

Diagonalization is the most pleasant outcome in matrix theory: choose a basis of eigenvectors and the matrix becomes a diagonal of scalars. But many important matrices are not diagonalizable, even over $\mathbb{C}$. The right response is not to abandon structure, but to ask what structure is still forced. Invariant subspaces, minimal polynomials, and Jordan form […]

If you had to choose one scalar invariant that shows up everywhere in linear algebra, rank would be a strong candidate. It is not merely a bookkeeping number attached \to a matrix. Rank controls what a linear map can do, what a system of equations can express, how solutions behave under perturbation, and how geometry […]

Linear algebra is full of statements that are perfectly true in the setting where most people first learn them: finite-dimensional vector spaces over a field, written in coordinates, with matrices you can row-reduce on a page. The trouble is that the mind quietly promotes “true in the standard setting” into “true in general,” and then […]

Orthogonality is one of those words that students recognize long before they understand. At first it means “perpendicular.” Later it becomes “dot product equals zero.” Then it becomes “independent directions.” Still later it becomes the engine behind least squares, projections, Fourier expansions, and stability of algorithms. The reason orthogonality keeps returning is that it is […]