Linear Algebra - Vector Space, Linear Map, Euclidean Space, Euclidean Vector, Rank, Determinant, Bra-Ket Notation, Dimension, Dual Spac (Paperback)

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 210. Chapters: Generalized eigenvector, Hilbert space, Vector space, Eigenvalues and eigenvectors, Matrix calculus, Determinant, Singular value decomposition, Euclidean vector, Seven-dimensional cross product, Jordan normal form, Bra-ket notation, Non-negative matrix factorization, System of linear equations, Theorems and definitions in linear algebra, Spectral theory, Newton's identities, Dual space, Eigendecomposition of a matrix, Euclidean subspace, Pseudovector, Rank (linear algebra), Split-complex number, Basis (linear algebra), Homogeneous coordinates, Dot product, Invertible matrix, Matrix Chernoff bound, Quadratic form, Linear map, Triangle inequality, Trace (linear algebra), Direct sum of modules, Norm (mathematics), Cauchy-Schwarz inequality, Entanglement-assisted stabilizer formalism, Orthogonality, Levi-Civita symbol, Projection (linear algebra), Cramer's rule, Change of basis, General linear group, Kernel (matrix), Gram-Schmidt process, Linear combination, Unit vector, Affine space, Kernel (algebra), Stabilizer code, Barycentric coordinate system (mathematics), Linear complementarity problem, Spinors in three dimensions, Vector-valued function, Squeeze mapping, Characteristic polynomial, Orientation (vector space), Linear independence, Linear functional. Excerpt: In linear algebra, for a matrix A, there may not always exist a full set of linearly independent eigenvectors that form a complete basis - a matrix may not be diagonalizable. This happens when the algebraic multiplicity of at least one eigenvalue is greater than its geometric multiplicity (the nullity of the matrix, or the dimension of its nullspace). In such cases, a generalized eigenvector of A is a nonzero vector v, which is associated with having algebraic multiplicity k 1, satisfying The set of all generalized eigenvectors for a given, together with the zero vector, form the generalized eigenspace for . Ordinary eigenvectors and eigenspaces are obtained for k=1. Generalized eigenvectors are needed to form a complete basis of a defective matrix, which is a matrix in which there are fewer linearly independent eigenvectors than eigenvalues (counting multiplicity). Over an algebraically closed field, the generalized eigenvectors do allow choosing a complete basis, as follows from the Jordan form of a matrix. In particular, suppose that an eigenvalue of a matrix A has an algebraic multiplicity m but fewer corresponding eigenvectors. We form a sequence of m eigenvectors and generalized eigenvectors that are linearly independent and satisfy for some coefficients, for . It follows that The vectors can always be chosen, but are not uniquely determined by the above relations. If the geometric multiplicity (dimension of the eigenspace) of is p, one can choose the first p vectors to be eigenvectors, but the remaining m p vectors are only generalized eigenvectors. Suppose Then there is one eigenvalue =1 with an algebraic multiplicity m of 2. There are several ways to see that there will be one generalized eigenvector necessary. Easiest is to notice that this matrix is in Jordan normal form, but is not diagonal, meaning that this is not a diagonalizable matrix. Since there is 1 superdiagonal entry, there will be one generalized eigenvector (or you could note