Invertible matrix

The inverted matrix is the transformation matrix for the basis we have chosen. The reaction in the LLNL database for Ca-clinoptilolite,... 

Diagonalize the matrix M, i.e., find a diagonal matrix D and an invertible matrix N such that M = NDN. Show that... 

All that is wanted here is the top row of the inverted matrix, since... 

MI Ml E S>. ° i= - U )2 del S, where s = su = - fik)(yu - Hfl) experimental errors interdependent with known covariance matrix are elements of the inverted matrix 53 1 experimental errors interdependent with unequal known variances experimental errors independent with equal (known or unknown) variances experimental errors interdependent with unknown covariance matrix no missing observations... 

The invertible matrix (J J) thus induces a change from the old coefficients (a, P, y) to new ones this is a map of 3-space with matrix... 

For samples of three constituents, one may use Cramers rule to solve the linear equations. If more components are involved, it is convenient to carry out the calculations with the inverted matrix AT. It follows from Eq. 5.1-20 that... 

Show, by use of Eqs. (A.1-12) and (A.1-10), that every symmetric invertible matrix A has a symmetric inverse. 

A set of simultaneous linear equations can also be solved by using matrices, as shown in Chapter 9. The solution matrix is obtained by multiplying the matrix of constants by the inverse of the matrix of coefficients. Applying this simple solution to the spectrophotometric data used above, the inverted matrix is obtained by selecting a 3R x 3C array of cells, entering the array formula... 

A matrix [/Il is said to be similar to a matrix [B] if there exists an invertible matrix [P] such that... 

Because the determinant of the matrix M is different from zero, there exists a complex invertible matrix T that diagonalize M, and X = T XMT is the diagonal matrix of complex eigenvalues, which can be found from the eigenvalue equation... 

The size of a matrix is the number of rows and columns hence, X in Eq. (A.4) is 4 x 5 (4 rows by 5 columns). The rank of a matrix is the number of linearly independent rows or columns (since the row and column rank of a matrix are the same). A matrix of size n x p that has rank less than p cannot be inverted (matrix inversion is discussed later). Similarly, a matrix of size n x n cannot be inverted if its rank is less than n. 

In algebra, a number multiplied by its inverse results in a value of 1. In matrix algebra, the inverse of a square matrix (denoted by a superscript T) multiplied by itself results in the identity matrix. In other words, the inverse of X is the matrix X-1 such that XX-1 = X-1X = I. Two matrices are said to be orthogonal or independent if XYT = I. The inverse of an orthogonal matrix is its transpose. Not all matrices can be inverted. However, one condition for inversion is that the matrix must be square. Sometimes an inverse to a matrix cannot be found, particularly if the matrix has a number of linearly dependent column. In such a case, a generalized estimate of inverted matrix can be estimated using a Moore Penrose inverse (denoted as superscript e.g., X-). 

In the method proposed by Broyden, Householder s formula was used to obtain the formula for the inverted matrix shown in step 6. The second term on the right-hand side of the expression in step 6 contains the correction proposed by Broyden. 

In many such cases we can find N by partitioning A = [AiAt], with Ax a square invertible matrix, and taking ... 

Thus one finds according to eq. (2.8) with the inverted matrix, which can be easily done using a commercial mathematical tool program package like MAPLE, using the following program lines ... 

Inversion of the symmetrical matrix as shown below or via triangularisation [572] gives the values Cjj, from which the coefficients a, b, and c of the regression model (eq. 114) are calculated by eqs. 121 — 123. matrix inverted matrix... 

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