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Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 42, No. 1, pp. 289-300 (2001)
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Principal Values and Principal Subspaces of Two Subspaces of Vector Spaces with Inner Product

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Ice B. Risteski, Kostadin G. Trencevski

2 Milepost Place #606, Toronto M4H 1C7, Canada, e-mail: iceristeski@hotmail.com; Institute of Mathematics, St. Cyril and Methodius University, P.O.Box 162, 91000 Skopje, Macedonia; e-mail: kostatre@iunona.pmf.ukim.edu.mk

**Abstract:** In this paper is studied the problem concerning the angle between two subspaces of arbitrary dimensions in Euclidean space $E_{n}$. It is proven that the angle between two subspaces is equal to the angle between their orthogonal subspaces. Using the eigenvalues and eigenvectors of corresponding matrix representations, there are introduced principal values and principal subspaces. Their geometrical interpretation is also given together with the canonical representation of the two subspaces. The canonical matrix for the two subspaces is introduced and its properties of duality are obtained. Here obtained results expand the classic results given in [H, K]. \item{[H]} Halmos, P. R.: * Finite Dimensional Vector Spaces*. 2nd ed. Van Nostrand Reinhold, New York 1958. \item{[K]} Kurepa, S.: * Finite Dimensional Vector Spaces and Applications*. Sveucilisna Naklada Liber, Zagreb 1979

**Keywords:** angles between subspaces, principal values, principal subspaces, principal directions

**Classification (MSC2000):** 15A03; 51N20

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