Linear Algebra

Linear algebra studies vector spaces, linear maps, matrices, duality, dimension, and spectral structure.

The largest field, special matrices, a third of the subject, covers circulant matrices, total positivity, the Perron–Frobenius theorem for irreducible nonnegative matrices and van der Waerden's lower bound \(n!/n^n\) for the permanent of a doubly stochastic \(n \times n\) matrix. Matrix decompositions and linear systems include the conjugate gradient method, whose energy-norm error after \(k\) steps is at most \(2\big((\sqrt{\kappa} - 1)/(\sqrt{\kappa} + 1)\big)^k\) times the initial one, \(\kappa\) being the condition number of the positive definite matrix. Eigenvalues and canonical forms include the Jordan normal form of a matrix whose characteristic polynomial splits, and positive definite matrices come with Hadamard's inequality and von Neumann's trace inequality.

• Linear maps and duality
• Eigenvalues and canonical forms
• Matrix decompositions and linear systems
• Determinants and matrix identities
• Positive definite matrices and matrix inequalities
• Bilinear and quadratic forms
• Special matrices

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