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The Diagonalization Theorem Sign Video

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Definition

Let A be an nXn matrix whose distinct eigenvalues are c_1, c_2, ... , c_k. The following statements are equivalent: (a) A is diagonalizable. (b) The union B of the bases of the eigenspaces of A contains n vectors. (c) The algebraic multiplicity of each eigenvalue equals its geometric multiplicity.

Source: Linear Algebra: A Modern Introduction, 3rd edition by David Poole (note-custom edition titled Matrix Algebra)

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> The Diagonalization Theorem

+ Linear Equation

+ Infinite Dimensional *

+ Binary Vector *

+ Linear Combination *

+ Dot Product *

+ Standard Unit Vectors *

+ Inconsistent *

+ Augmented Matrix *

+ Coefficient Matrix *

+ Row Echelon Form *

+ Elementary Row Operations *

+ Back Substitution *

+ Leading Entry *

+ Row Reduction *

+ Gaussian Elimination *

+ Leading Variable *

+ Free Variable *

+ The Rank Theorem *

+ Gauss Jordan Elimination *

+ Reduced Row Echelon Form *

+ Linearly Dependent *

+ Linearly Independent *

+ Strictly Diagonal Dominant *

+ Diagonal Entries *

+ Square Matrix *

+ Diagonal Matrix *

+ Scalar Matrix *

+ Zero Matrix *

+ Column Matrix *

+ Elementary Matrix *

+ Matrix Factorization *

+ Column Space *

+ Standard Basis *

+ Basis Theorem *

+ Inverse Transformation *

+ Upper Triangular *

+ Lower Triangular *

+ Stochastic Matrix *

+ Identity Transformation *

+ Linear Transformation *

+ Eigenvector *

+ Cofactor Matrix *

+ Cramer's Rule *

+ Characterstic Polynomial *

+ Dominant Eigenvalue *

+ Algebraic Multiplicity *

+ Laplace Expansion Theorem *

+ Characteristic Equation *

+ Geometric Multiplicity *

+ Similar Matrices *

- The Diagonalization Theorem *

+ Orthogonal Set *

+ Orthogonal Basis *

+ Orthonormal Set *

+ Orthonormal Basis *

+ Orthogonal Matrix *

+ Diagonalizable *

+ Orthogonal Complement *

+ Qr Factorization *

+ Orthogonally Diagonalizable *

+ The Spectral Theorem *

+ Fibonacci Type Sequence *

+ Lucas Sequence *

+ Vector Space *

+ Zero Subspace *

+ Orthogonal Decomposition Theorem *

+ Trivial Subspace *

+ Magic Square *

+ Zero Transformation *

+ Fund Thm Of Invertible Matrices *

+ Finite Dimensional *

+ Characteristic Polynomial *

+ Column Vector *

+ Null Vector *

+ Conjugate Matrix *

+ Nonsingular Matrix *

+ Singular Matrix *

+ Gram Schmidt Orthonormalization *

+ System Of Linear Equations

* video needed

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