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Geometric mean of complex numbers

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Geometric mean of positive definite matrices

The geometric mean can be extended to positive-definite matrices, and even to accretive matrices, i.e., matrices with a strictly positive Hermitian part.

GA^{-1}G=B.

G^{-1}={\frac {2}{\pi }}\int _{0}^{\infty }\left(tA+t^{-1}B\right){\frac {dt}{t}}.

A\#B=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2}=B^{1/2}(B^{-1/2}AB^{-1/2})^{1/2}B^{1/2}.

It is the midpoint on the geodesic connecting $A$ and $B$ on the smooth manifold of positive definite matrices.

A collection of references needed, also exploring different options for how citations work in Wiki. A sentence.^[1] Another sentence.^[2] A third sentence.^[3] A forth sentence.^[3]^{: 233–254} A fifth sentence.^[3]^: 233-254 A sixth sentence.^[3]^{: chapter 6} A seventh sentence.^[4]

^ Lawson, Jimmie D.; Lim, Yongdo (2001). "The Geometric Mean, Matrices, Metrics, and More". The American Mathematical Monthly. 108 (9): 797–812. doi:10.2307/2695553.
^ Kubo, Fumio; Ando, Tsuyoshi (1980). "Means of Positive Linear Operators". Mathematische Annalen. 246 (3): 205–224. doi:10.1007/BF01371042.
^ ^a ^b ^c ^d Bhatia, Rajendra (2007). Positive definite matrices. Princeton University Press. ISBN 978-0-691-12918-1.
^ Drury, Stephen (2015). "Principal powers of matrices with positive definite real part". Linear and Multilinear Algebra. 63 (2): 296–301. doi:10.1080/03081087.2013.865732.

[Lawson2001-1] Lawson, Jimmie D.; Lim, Yongdo (2001). "The Geometric Mean, Matrices, Metrics, and More". The American Mathematical Monthly. 108 (9): 797–812. doi:10.2307/2695553.

[Kubo1980-2] Kubo, Fumio; Ando, Tsuyoshi (1980). "Means of Positive Linear Operators". Mathematische Annalen. 246 (3): 205–224. doi:10.1007/BF01371042.

[Bhatia2007-3] Bhatia, Rajendra (2007). Positive definite matrices. Princeton University Press. ISBN 978-0-691-12918-1.

[Drury2015-4] Drury, Stephen (2015). "Principal powers of matrices with positive definite real part". Linear and Multilinear Algebra. 63 (2): 296–301. doi:10.1080/03081087.2013.865732.

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