Root characteristics

Leak or spill source Characteristics Root causes Preventive measures... 

Banachiewicz method, 67 characteristic roots, 67 characteristic vectors, 67 Cholesky method, 67 Danilevskii method, 74 deflation, 71 derogatory form, 73 "equations of motion, 418 Givens method, 75 Hessenberg form, 73 Hessenberg method, 75 Householder method, 75 Jacobi method, 71 Krylov method, 73 Lanczos form, 78 method of modification, 67 method of relaxation, 62 method of successive displacements,... 

Historically, gas lubrication theory was developed from the classical liquid lubrication equation—Re5molds equation [4]. The first gas lubrication equation was derived by Harrison [5] in 1913, taking the compressibility of gases into account. Because the classical gas lubrication equation is based on the Navier-Stokes equation, it does not incorporate some gas flow characteristics rooted in the rarefaction effects of dilute gases. As early as 1959, Brunner s experiment [6] showed that the classical gas lubrication equation was... 

An eigenvalue or characteristic root of a symmetric matrix A of dimension p is a root of the characteristic equation ... 

By way of example we construct a positive semi-definite matrix A of dimensions 2x2 from which we propose to determine the characteristic roots. The square matrix A is derived as the product of a rectangular matrix X with its transpose in order to ensure symmetry and positive semi-definitiveness ... 

Sacan MT, Balcioglu IA. 1996. Prediction of the soil sorption coefficient of organic pollutants by the characteristic root index model. Chemosphere 32 1993-2001. 

Gershgorin theorem (Marcus and Mine, 1992, p. 146) The characteristic roots of A lie in the closed region of the z-plane... 

The trace of the inverse equals the sum of the characteristic roots of the inverse, which are the reciprocals of the characteristic roots of X X. 

Suppose that the regression model is y = fi+e, where t lias a zero mean, constant variance, and equal correlation p across observations. Then Cov ty.ty = crp Vi -t- j. Prove that the least squares estimator of ju is inconsistent. Find the characteristic roots of Q and show that Condition 2. after Theorem 10.2 is violated. 

To find the determinant, use the product of the characteristic roots. Note first that... 

The determinant of a matrix equals the product of its characteristic roots, so the log determinant equals the sum of the logs of the roots. The characteristic roots of the matrix above remain to be detennined. As shown in the exercise, T-1 of the T roots equal 1. Therefore, the logs of these roots are zero, so the log-detenninant equals the log of the remaining root. It remains only to find the other characteristic root. Premultiply the result (o 2/a 2)ii c = (A-1 )c by i to obtain (a 2/a 2)i ii c = (A-l)i c. 

We would require that all three characteristic roots have modulus less than one. An intuitive guess that the diagonal element greater than one would preclude this would be correct. The roots are the solutions to... 

Use the spectral decomposition to write A as CAC where A is the diagonal matrix of characteristic roots. Then, the Klh power of A is CAaC. Sufficiency is obvious. Also, since if some X is greater than one, Ak must explode, the condition is necessary as well. 

The determinant of A is nonzero if A is nonsingular, so the solutions to the two detenninantal equations must be the same. B A is the inverse of A 1B, so its characteristic roots must be the reciprocals of those of A" B. There might seem to be a problem here since these two matrices need not be symmetric, so the roots could be complex. But, for the application noted, both A and B are symmetric and positive definite. As such, it can be shown (see Section 16.5.2d) that the solution is the same as that of a third determinantal equation involving a symmetric matrix. 

Suppose that A is an nxn matrix of the form A = (l-pl) + pii, where i is a column of Is and 0 < p < 1. Write out the format of A explicitly for n = 4. Find all of the characteristic roots and vectors of A. (Hint There are only two distinct characteristic roots, which occur with multiplicity 1 and n-1. Every c of a certain type is a characteristic vector of A.) For an application which uses a matrix of this type, see Section 14.5 on the random effects model. 

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