Symmetric positive definite matrix

If A is a symmetric positive definite matrix then we obtain that all eigenvalues are positive. As we have seen, this occurs when all columns (or rows) of the matrix A are linearly independent. Conversely, a linear dependence in the columns (or rows) of A will produce a zero eigenvalue. More generally, if A is symmetric and positive semi-definite of rank r[Pg.32]

A symmetric positive definite matrix, A, can also be written as A = UL, where U is an upper triangular matrix and L U. This is not the Cholesky decomposition, however. Obtain this decomposition of the matrix in Exercise 9. 

Preconditioning is a technique which improves the condition number of a matrix and thereby increases the convergence rate of Krylov subspace methods. Thus, if the preconditioner A4 is a symmetric, positive definite matrix, the original problem Ax = b can be solved indirectly by solving M Ax = M h. The the residual can then be written as ... 

Calculate the canonical partition function for Qxp(—Pq Kq/2) for the case where q e Q = where is a symmetric positive definite matrix. 

Proposition 8.3 Assume A to be a symmetric positive definite matrix with distinct eigenvalues, and let Ck, Dk be defined as in Lemma 8.1, then the 2Nc vectors Ci,Di, C2,D2., form a linearly independent set on T). 

It is frequently effective to use block representation of parallel algorithms. For instance, a parallel version of the nested dissection algorithm of Section VIII.C for a symmetric positive-definite matrix A may rely on the following recursive factorization of the matrix Ao = PAP, where P is the permutation matrix that defines the elimination order (compare Sections III.G-I) ... 

Two vectors, pj and pj, are reciprocally conjugate with respect to a symmetric positive definite matrix, A, if... 

By replacing the identity matrix with a symmetric positive definite matrix G, the following relation takes place ... 

Our companion book (Buzzi-Ferraris and Manenti, 2010a) discusses the updating procedure of L and D that factorizes a symmetric positive definite matrix to which a rank-1 ss -type matrix is added. Obviously, since two of such matrices are added sequentially here, this procedure should be repeated twice. It is important to know that the new factorization requires 0(wy) calculations. 

The Cholesky method (Buzzi-Ferraris and Manenti, 2010a), which requires a symmetric positive definite matrix. 

The inverse of the Hessian matrix is approximated by a symmetric positive definite matrix that is constructed iteratively. To that end, the Scilab function optim () [10] uses the Broyden, Fletcher, Goldfarb and Shanno (BFGS) update [11], Alternatively, the Levenberg-Marquardt algorithm implemented in the Scilab function IsqrsolveO may be used. 

The coefficients L m are called the mobility matrix, and may be obtained using hydrodynamics. It can be proved that L , is a symmetric positive definite matrix ... 

Let us now have a general set of linear equations (constraints) (7.1.1), with the regularity condition (7.1.4), and with the partition of variables (7.1.9) into measured (x R ) and unmeasured (y e R ). We thus also know the set (7.3.4) of vectors x obeying the condition of solvability, see (7.3.2). Let x" be the vector of actually measured values. Suppose we know the covariance matrix F of measurement errors it is an / x / symmetric positive definite matrix. If the matrix is diagonal (errors uncorrelated) thus... 

We now consider the origin of the excellent performance of the conjugate gradient method for a quadratic cost function, defined in terms of a symmetric, positive-definite matrix A and a vector b,... 

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