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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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  • Example of vector-scalar multiplication in R^2#23794
  • Example of writing a given vector in R^3 as a linear combination of given vectors#23803
  • Examples#23886
  • Examples of vector spaces#24095
  • Factorization of matrices#23622
  • Finite-dimensional Fitting decomposition into generalized kernel and range#24061
  • Finite-dimensional vector spaces of equal dimension over the same field are isomorphic#24005
  • F^n is a vector space.#24099
  • Formula for computing the least squares solution to a linear system#23949
  • Formula for computing the least squares solution to a linear system.#23948
  • Formula for the coordinates of a vector with respect to an orthogonal/orthonormal basis.#23933
  • Formula for the coordinates of the projection of a vector onto a subspace#23944
  • Formula for the determinant of a 2-by-2 matrix.#23742
  • Formula for the determinant of a 3-by-3 matrix.#23743
  • Formula for the inverse of a 2-by-2 matrix.#23521
  • Formula for the least-squares linear fit to 2-dimensional data#23952
  • Formula for the (orthogonal) projection of one vector onto another vector#23939
  • Formula for the spectral decomposition for a symmetric matrix#23576
  • For n-by-n invertible matrices A and B#23523
  • Gaussian elimination as a method to solve a linear system#23421
  • Gauss-Jordan procedure to put a matrix into reduced row echelon form#23540
  • Geometric description of span of a set of vectors in R^n (or C^n)#23806
  • Geometric picture of a 2-by-2 linear system#23428
  • Geometric picture of a 3-by-3 linear system#23430
  • Geometric picture of the solution set of a linear equation in 3 unknowns#23429
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  • Aspen Summer 2016 - Entanglement Matters#1213
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