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An Lower-diagonal-upper (LDU) decomposition is a decomposition of the form where D is a diagonal matrix, and L and U are unitriangular matrices, meaning that all the entries on the diagonals of L and U are one. Above we required that A be a square matrix, but these decompositions can all be generalized to rectangular matrices as well.

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The unit circle and its transformation by the matrix A. The vectors are the eigenvectors of A. We can see the unit circle in dark blue, the non scaled eigenvectors in light blue, the transformed unit circle in green and the scaled eigenvectors in yellow. It is worth noting that the eigenvectors are orthogonal here because the matrix is symmetric.

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An Lower-diagonal-upper (LDU) decomposition is a decomposition of the form where D is a diagonal matrix, and L and U are unitriangular matrices, meaning that all the entries on the diagonals of L and U are one. Above we required that A be a square matrix, but these decompositions can all be generalized to rectangular matrices as well.

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A (non-zero) vector v of dimension N is an eigenvector of a square N × N matrix A if it satisfies the linear equation = where λ is a scalar, termed the eigenvalue corresponding to v.That is, the eigenvectors are the vectors that the linear transformation A merely elongates or shrinks, and the amount that they elongate/shrink by is the eigenvalue. The above equation is called the eigenvalue

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Eigenvalues and Eigenvectors. A vector is a quantity which has both magnitude and direction. The general effect of matrix A on the vectors in x is a combination of rotation and stretching. For example, it changes both the direction and magnitude of the vector x1 to give the transformed vector t1.However, for vector x2 only the magnitude changes after transformation.

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Specifically, the singular value decomposition of an × complex matrix M is a factorization of the form ∗, where U is an × complex unitary matrix, is an × rectangular diagonal matrix with non-negative real numbers on the diagonal, and V is an × complex unitary matrix. If M is real, U and = ∗ are real orthogonal matrices.. The diagonal entries = of are known as the singular values of M.

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This document is copyrighted by DILO Company, Inc., 11642 Pyramid Drive, Odessa, FL 33556 - USA. Any use, duplication in whole or in part without the express written permission of DILO is strictly prohibited. DILO Company, Inc. SF 6 Maintenance Equipment West Pasco Industrial Park 11642 Pyramid Drive Odessa, FL 33556 U.S.A Tel.: (727) 376-5593

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Vector decomposition is the general process of breaking one vector into two or more vectors that add up to the original vector. The component vectors into which the original vector is decomposed are chosen based on specific details of the problem at hand. Note that the coordinate unit vectors are orthonormal.

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