Every Hermitian positive-definite matrix ( and thus also every real-valued symmetric positive-definite matrix ) has a unique Cholesky decomposition.
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For every pixel ( i, j ) in the image, the structure tensor J is a symmetric and positive semi-definite matrix.
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The diffusion tensor, a 3 & times; 3 symmetric positive-definite matrix, offers a straightforward solution to both of these goals.
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*PM : square root of positive definite matrix, id = 7067-- WP guess : square root of positive definite matrix-- Status:
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*PM : square root of positive definite matrix, id = 7067-- WP guess : square root of positive definite matrix-- Status:
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Every real non-singular matrix can be uniquely factored as the product of an orthogonal matrix and a symmetric positive definite matrix, which is called a polar decomposition.
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Many of these methods are only applicable to certain types of equations, for example the Cholesky factorization and conjugate gradient will only work if is a positive definite matrix.
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Is negative definite for some positive definite matrix M = M ^ { T } . ( The relevant Lyapunov function is V ( x ) = x ^ TMx .)
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Mathematically this means that the result of subtracting the expected squared error ( which is not usually known ) from M is a semi-definite or positive-definite matrix.
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More generally, when " D " is a symmetric positive definite matrix, the equation describes anisotropic diffusion, which is written ( for three dimensional diffusion ) as:
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