Matrices symmetric

A symmetric matrix is a matrix equal to its transpose. A symmetric matrix must be a square matrix, for example. 

A diagonal matrix is a symmetric matrix with all of its entries equal to zero, except possibly the values of the diagonal, for example. 

A square matrix is one in which the number of columns is equal to the number of rows. An important type of square matrix which arises quite often in the Unite element method is a symmetric matrix. Such matrices possess the property that Qij = aji- An example of such a matrix is given below  

Symmetric matrices are square matrices that are identical to their transpose. They are invariant to an inflection at their main diagonal, i.e. invariant to the interchange of row and column index. In the former Matlab example both the 2x2 matrix Y1 and the 3x3 matrix Y2 are symmetric. 

Diagonal matrices are handy when individual rows or columns of a matrix are to be multiplied by different scalar factors si...sn. One typical example is the normalisation of B so that the square root of the sum of all squared elements in, for example, each row of B becomes one, i.e. unity length of each row vector. 

Note that the Matlab command sum(B.A2,2) performs a row-wise addition of all squared elements of B. For a column-wise summation sum (B. A2,1) or just sum(B. A2) could be used. 

Another common task is to normalise in such a way that the maximum value in each e.g. column of B, becomes one. 

Note that the command max(B,l) or simply max(B) finds the column-wise maxima, max (B, 2) the row-wise maxima of B. 

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Subroutine SYMINV. This subroutine inverts a symmetric matrix. 

This definition implies that the direction of the vector d was taken into account and that the coefficient c(i,j,fd)sndc(j,i,f,-d) may be different. So, the symmetrical matrix consists of a two step operation ... 

Given a graph G with n vertices, the adjacency matrix A is a square nxn symmetric matrix (Eq. 9 ). 

The distance matrix D of a graph G with n vertices is a square n x n symmetric matrix as represented by Eq. (13), where is the distance between the vertices Vi and Vj in the graph (i.e., the number of edges on the shortest path). 

The transpose of a square matrix is, of course, another square matrix. The transpose of a symmetric matrix is itself. One particularly important transpose matrix is the adjoint natris, adJA, which is the transpose matrix of cofactors. For example, the matrix of cofactors ul liie 3x3 matrix... 

A tircial solution to this equation is x = 0. For a non-trivial solution, we require that the deterniinant A - AI equals zero. One way to determine the eigenvalues and their associated eigenvectors is thus to expend the determinant to give a polynomial equation in A. Ko." our 3x3 symmetric matrix this gives ... 

Ihe metric matrix G is a square symmetric matrix. A general property of such matrices that they can be decomposed as follows ... 

The Hermetian conjugate plays the same role for complex matrices that the symmetric matrix plays for real matrices. 

The form of the symmetric matrix of coefficients in Eq. 3-20 for the normal equations of the quadratic is very regular, suggesting a simple expansion to higher-degree equations. The coefficient matrix for a cubic fitting equation is a 4 x 4... 

The force constants k 2 and k2 are the off-diagonal elements of the matrix. If they are zero, the oscillators are uncoupled, but even if they are not zero, the K matrix takes the simple fomi of a symmetrical matrix because ki2 = k2. The matrix is symmetrical even though may not be equal to k22-... 

As F is a symmetric matrix, there exists an orthogonal transformation that diagonalizes F ... 

The determination of some of the eigenvalues and eigenvectors of a large real symmetric matrix has a long history in numerical science. Of particular interest in the normal mode... 

The matrices [G] and [F] are column matrices with row numbers n and k, respectively. The matrix solution is simplified by special properties of the symmetric matrix and because the resulting values of G occur in complex conjugate pairs. In general, we may write... 

Symmetric matrix This is a square matrix where eiements a ... 

H is a real symmetric matrix, and its eigenvalues are therefore real. Its eigenvectors are referred to as the principal axes of curvature . 

We now consider two special cases of the symmetric matrix A] namely, (1) the entries of A are allowed to take on only integer values, and (2) the entries of A are all binary valued. 

Corollary 1 [goles87a] If A is an n x n integer-valued symmetric matrix, then the transient length of the generalized threshold rule defined in equation 5.121 is bounded by... 

This method has very little other than its simplicity to recommend it in the form just described. But when a binary base is used, the corresponding procedure is to bisect the interval successively. Each bisection determines one additional binary digit to the approximation, it requires only the evaluation of the function, and the method is often efficient and accurate. The principle is used by Givens (Section 2.3) in finding the roots of a tridiagonal symmetric matrix. 

Proof.—There exists a unitary skew symmetric matrix C with the property that... 

Difference equations with a symmetric matrix are typical in numerical solution of boundary-value problems associated with self-adjoint differential equations of second order. In what follows we will show that the condition Bi = is necessary and sufficient for the operator [yj] be self-adjoint. As can readily be observed, any difference equation of the form... 

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