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  • Matrix diagonalization

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  • Definition of matrix diagonalization
  • Definition of diagonalizable matrix
  • An n-by-n matrix is diagonalizable if and only if it has n linearly independent eigenvectors.
  • An n-by-n matrix is diagonalizable if and only if the sum of the dimensions of the eigenspaces equals n.
  • Diagonalizability requires splitting and equality of geometric and algebraic multiplicities
  • A diagonalizable matrix is diagonalized by a matrix having the eigenvectors as columns.
  • An n-by-n matrix is diagonalizable if and only if the union of the basis vectors for the eigenspaces is a basis for R^n (or C^n).
  • An n-by-n matrix with n distinct eigenvalues is diagonalizable.
  • Real block form and complex diagonalization for a 2-by-2 matrix
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  1. An n-by-n matrix is diagonalizable if and only if the union of the basis vectors for the eigenspaces is a basis for R^n (or C^n).

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  • Geometric properties of linear transformations#23851
  • Geometric properties of linear transformations on R^2#23853
  • Geometric properties of R^n (or C^n)#23795
  • Hermitian matrices#23596
  • Hermitian matrices have real eigenvalues.#23711
  • Homogeneous linear systems are consistent.#23434
  • If a matrix has both a left and a right inverse#23519
  • If a set of vectors contains the 0 vector#23825
  • If a set of vectors in R^n (or C^n) contains more than n elements#23827
  • If a square matrix has a one-sided inverse#23520
  • If a vector space has dimension n#23838
  • If the product of a vector and a scalar is 0#23962
  • If two matrices have equal products with all vectors#23484
  • Ill-conditioning is measured by relative distance to singularity#23749
  • Inner products#24087
  • Inner products in coordinate spaces#23894
  • Invariant direct sums give block diagonal operator matrices#24053
  • Inverse#23514
  • Isomorphic vector spaces have the same dimension.#24004
  • Isomorphism#24000
  • Least squares#23945
  • Linear algebra#23410
  • Linear Algebra content audit and completionArchived#29496
  • Linear Algebra practice problems with worked solutions#37116
  • Linear Algebra study guide, conventions and references#37114
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