By Frank Nielsen, Rajendra Bhatia
-Presents advances in matrix and tensor facts processing within the area of sign, photograph and knowledge processing
-Written by way of specialists within the components of theoretical arithmetic or engineering sciences
-Discusses power purposes in sensor and cognitive platforms engineering
This ebook is an end result of the Indo-French Workshop on Matrix info Geometries (MIG): functions in Sensor and Cognitive platforms Engineering, which was once held in Ecole Polytechnique and Thales study and know-how heart, Palaiseau, France, in February 23-25, 2011. The workshop was once generously funded by means of the Indo-French Centre for the merchandising of complicated study (IFCPAR). throughout the occasion, 22 popular invited french or indian audio system gave lectures on their components of craftsmanship in the box of matrix research or processing. From those talks, a complete of 17 unique contribution or cutting-edge chapters were assembled during this quantity. All articles have been completely peer-reviewed and greater, in accordance with the feedback of the overseas referees. The 17 contributions provided are geared up in 3 components: (1) cutting-edge surveys & unique matrix conception paintings, (2) complicated matrix concept for radar processing, and (3) Matrix-based sign processing purposes.
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Bm ). −1 −1 8. Self-duality. G(A1 , . . , Am ) = G(A−1 1 , . . , Am ) . 9. Determinant identity. det G(A1 , . . , Am ) = (det A1 · det A2 . . det Am )1/m . 10. Arithmetic-geometric-harmonic mean inequality. ⎛ ⎝1 m m j=1 ⎞−1 ⎠ A−1 j 1 ≤ G(A1 , . . , Am ) ≤ m m Aj. j=1 When m = 2, the binary mean G(A1 , A2 ) = A1 # A2 satisfies all these conditions. For m > 2 the ALM mean is defined inductively. Suppose a geometric mean G # has been defined for (m − 1) tuples. Then given an m-tuple A = (A1 , .
Let P(n) be the set of n × n positive matrices. Imitating the five conditions above we could say that a matrix mean is a map M : P(n) × P(n) → P(n) that satisfies the following conditions: (i) M(A, B) = M(B, A). (ii) If A ≤ B, then A ≤ M(A, B) ≤ B. (iii) M(X ∗ AX, X ∗ B X ) = X ∗ M(A, B)X, for all nonsingular matrices X . (Here X ∗ is the conjugate transpose of X ). (iv) A ≤ A ⇒ M(A, B) ≤ M(A , B). (v) M is continuous. The arithmetic and the harmonic means defined, respectively as 2 The Riemannian Mean of Positive Matrices A+B , 2 37 −1 A−1 + B −1 2 do have the five properties listed above.
Appl. 29, 328–347 (2007) 32 J. Angulo 7. : On the Löwner, minus, and start partial orderings of nonnegative definite matrices and their squares. Linear Algebra Appl. 151, 135–141 (1991) 8. : New foundation of radar doppler signal processing based on advanced differential geometry of symmetric spaces: doppler matrix CFAR and radar application. In: Proceedings of International Radar Conference, Bordeaux, France (2009) 9. : Geometric radar processing based on Fréchet distance: information geometry versus optimal transport theory.