Positive Definiteness . . . Eigenvalues of Tensors Z-eigenvalue Methods . . . More Study on E- . . . Lim’s Exploration Further Discussion

Eigenvalues of Tensors and Their Applications

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by L IQUN Q I Department of Mathematics City University of Hong Kong

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Positive Definiteness . . .

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Eigenvalues of Tensors Z-eigenvalue Methods . . . More Study on E- . . . Lim’s Exploration

 Positive Definiteness of Multivariate Forms

Further Discussion

 Eigenvalues of Tensors Home Page

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 Z-Eigenvalue Methods and The Best Rank-One Approximation

 More Study on E-Eigenvalues and Z-eigenvalues

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 Lim’s Exploration

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Positive Definiteness . . . Eigenvalues of Tensors Z-eigenvalue Methods . . . More Study on E- . . . Lim’s Exploration Further Discussion

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1. Positive Definiteness of Multivariate Forms Recently, I defined eigenvalues and eigenvectors of a real supersymmetric tensor, and explored their practical applications in determining positive definiteness of an even degree multivariate form, and finding the best rank-one approximation to a supersymmetric tensor. This work extended the classical concept of eigenvalues of square matrices, and has potential applications in mechanics and physics as well as the classification of hypersurfaces and the study of hypergraphs.

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Positive Definiteness . . . Eigenvalues of Tensors

1.1.

Z-eigenvalue Methods . . .

Quadratic Forms

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Consider a quadratic form f (x) ≡ xT Ax =

Lim’s Exploration

n X

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aij xi xj ,

i,j=1

where x = (x1 , · · · , xn )T ∈ 0 for all x ∈