Norm of a matrix

The euclidean norm of a matrix considered as a vector in m2-space is a matrix norm that is consistent with the euclidean vector norm. This is perhaps the matrix norm that occurs most frequently in the literature. But the euclidean norm of I is n112 > 1 when n > 1, hence it is not a sup. In fact,... 

Note that the expression on the right hand side of the inequality (A.27) is called the Frobenius norm of a matrix, HA1 ... 

The Zqo norm of a matrix is the largest value, taken over the rows of the matrix, of the sum of the absolute values of the entries in that row. 

II. Ilf represents the F norm of a matrix, which is the square root of the sum of the squares of all elements. The matrix E is the error matrix of the second-order measurements. 

The norm of a matrix is said to be compatible with a given vector norm provided Eq. (A-29) is satisfied for any matrix A and any conformable vector X. The following procedure is employed for the construction of the matrix norm such that it is compatible with a given vector norm. First, observe that the lengths of all vectors in any ri-dimensional vector space span the set of all real numbers, and that the process of normalization of each vector throws the lengths of all... 

Condition (condition number) Product of the norms of a matrix and of its inverse condition of the coefficient matrix characterizes the sensitivity of the solution of the linear system to input errors. 

Use the facts that the trace, the determinant, and the norm of a matrix are invariant under an orthogonal transformation to find the eigenvalues of the following matrices ... 

Several norms of a matrix are defined in the existing literature. The Frobenius norm is defined as ... 

This method requires solution of sets of linear equations until the functions are zero to some tolerance or the changes of the solution between iterations is small enough. Convergence is guaranteed provided the norm of the matrix A is bounded, F(x) is Bounded for the initial guess, and the second derivative of F(x) with respect to all variables is bounded. See Refs. 106 and 155. 

The condition number of a matrix A is intimately connected with the sensitivity of the solution of the linear system of equations A x = b. When solving this equation, the error in the solution can be magnified by an amount as large as cortd A) times the norm of the error in A and b due to the presence of the error in the data. 

Alternatively, Matlab s built-in function norm can be used to determine normalisation coefficients and perform the same task. An example for column-wise normalisation of a matrix X with orthogonal columns is given below. It is worthwhile to compare X with equation (2.15) the subspace command can be used to determine the angle between the vectors (in rad) and reconfirm orthogonality. ... 

Here RA is the response of the analyte at unit concentration, c E is a matrix of expected, or estimated, errors and F is the Froebus norm, or root sum of the squared elements, of a matrix. It should be noted that while the NAS is a matrix quantity, selectivity (SEL), sensitivity (SEN), and signal-to-noise (S/N) are all vector quantities. The limit of detection and the limit of quantitation can also be determined via any accepted univariate definition by substituting NAS P for the analyte signal and E P for the error value. 

The norm function of a matrix is often useful and consists of the square root of the sum of squares, so in our example norm (W) equals 12.0419. This can be useful when scaling data, especially for vectors. Note that if Y is a row vector, then sqrt (Y Y ) is die same as norm(Y). 

The definition above of a matrix norm is not directly evaluable in finite time. However, it is possible to determine the value of each of the norms from the elements of a matrix without working through all possible vectors. 

We can get some handle on this value by noting that any norm is an upper bound on the spectral radius of a matrix. If, during the tending of to to oo, it is found that the matrix given by one of the product sequences has a norm equal to its spectral radius, then that will be the joint spectral radius of the A and B. 

The condition on a non-negative basis function is essentially a condition on the Zoo norm of the matrix of basis functions, introduced in the last chapter, maxt Ei b(t — i) < 1 which, because Sib(t — i) = 1, cannot be less than 1. 

In an analogous manner, the norm of a square matrix A is defined as a nonnegative number A satisfying the following conditions ... 

First it will be shown that for any real nonsingular matrix A of order n, there exists a particular vector Xx such that jXj = 1 and flAXj = A. The matrix norm so constructed is said to be subordinate to the given vector norm. Then it will be shown that the matrix norm so constructed satisfies the four conditions required of a matrix norm as well as the compatibility relationship. Only the matrix norm subordinate to the euclidean vector norm X , given by Eq. (A-26) is considered. Now consider the vector AX, whose vector norm as given by Eq. (A-26), is... 

Frobhenius norm due to convenience and geometric interpretation. The condition number of a matrix can be calculated in the following way [18] ... 

A tensor of rank 2 can be represented by a 3 x 3 matrix (three-dimensional space 9i ). However, this must not be confused with a matrix which describes a change in coordinate system. A physical magnitude is invariant with respect to a change in coordinates. In particular, the norm of a vector ( E, J p) and the scalar product of two vectors (E-J) = E J = J E must be independent of the chosen coordinate system. 

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