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− | + | == Diagonalization == | |
+ | Diagonalization is the process of tranforming a square matrix $A$ to the diagonal form | ||
+ | |||
+ | Diagonal Form | ||
+ | * $T$ is the ''diagonal form'' of $A$ if | ||
+ | * $T$ is Diagonal and | ||
+ | * there exists $X$ such that $T = X^{-1} A X$ | ||
+ | * $A$ and $T$ are [[Similar Matrices|similar]] | ||
+ | * so $A$ and $T$ share the same [[Eigenvalues and Eigenvectors|eigenvalues]] | ||
+ | |||
+ | |||
+ | Non-defectiveness | ||
+ | * $A$ is ''non defective'' $\iff$ there exists non singular $X$ s.t. | ||
+ | * $X^{-1} A X = \text{diag}(\lambda_1, ..., \lambda_n)$ | ||
+ | * i.e. there exists a similarity tranformation $X$ such that the results is a diagonal matrix with eigenvalues on the diagonal | ||
+ | |||
+ | |||
+ | == [[Eigendecomposition]] == | ||
+ | Eigendecomposition decomposes a symmetric matric $A$ as | ||
+ | * $\Lambda = S^{T} A S$ | ||
+ | * where $\Lambda = \text{diag}(\lambda_1, ..., \lambda_n)$ | ||
+ | * and $S$ has eigenvectors on the diagonal | ||
+ | * so the similarity transformation matrix $S$ here is [[Orthogonal Matrices|orthogonal]] | ||
+ | |||
+ | |||
+ | == Sources == | ||
+ | * [[Linear Algebra MIT 18.06 (OCW)]] | ||
+ | * [[Matrix Computations (book)]] | ||
+ | |||
+ | [[Category:Linear Algebra]] |
Diagonalization is the process of tranforming a square matrix $A$ to the diagonal form
Diagonal Form
Non-defectiveness
Eigendecomposition decomposes a symmetric matric $A$ as