matrixfacts - Selected Properties of Matrices,...

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Selected Properties of Matrices, Determinants, Solutions of Linear Equations, Eigenvalues, and Eigenvectors Last Updated: October 24, 2007 In what follows, A is an n by n real matrix unless otherwise stated. The inverse of A is denoted by A -1 and its determinant is | A |. Column vectors are denoted by lower case bold letters; e.g. x , y , and z and can be assumed to be of length n unless otherwise stated. The transpose of a matrix is A T and a row vector is x T . 1.0 Solution of Linear Equations 1.1 Equivalent Statements Regarding Ax = b a. A is nonsingular b. Ax = b has a unique nonzero solution c. Ax = 0 has only x = 0 as its unique solution d. | A | 0 e. A -1 exists and A -1 A = AA -1 = I (the identity matrix) f. Gaussian elimination works (with row swaps when necessary to avoid a zero pivot). g. Row operations on A do not produce a zero row 1.2 Matrix Decompositions a. A = L U = LU = L DU if the upper-left principal submatrices are nonsingular where L is unit lower, U is unit upper and D is diagonal. b. If A is symmetric and positive definite, then A = L D 1/2 D 1/2 U = LL T where D 1/2 is the diagonal matrix whose entries are the square root of the diagonals of D and L = L D 1/2 = ( D 1/2 U ) T 1.3. Solving Ax = b by Iteration (Jacobi (JA), Gauss-Seidel
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matrixfacts - Selected Properties of Matrices,...

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