Rank of matrix

Definition The rank of a matrix is the number of independent rows or columns in the matrix. 

When performing Gaussian elimination, the rank becomes evident as rows are eliminated. The rank can be deduced from the triangular matrix by observing the number of nonzero rows and columns. 

Having defined rank and knowing at least one way to compute it, the following statements can be made about the n x n linear system Ax = b  

Consistency A linear system is consistent if rank(A) = rank([A I b ). That is, both A and the augmented matrix [A I fc] have the same rank. 

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The rank of matrix M is 7. As the system is rank deficient, it admits a decomposition into two subsystems, one estimable and the other nonestimable. To determine which variables are observable, the column echelon form of M is obtained and T l... 

Regarding the unmeasured process variables, it was shown in Chapter 4 that the rank of matrix Ru is equal to 6 this means that at least one of the unmeasured variables is indeterminable. The remaining ones can be written in terms of it, as Eq. (5.27) indicates. In this case, from the orthogonal transformation, the subsets of u are defined as... 

Rank of matrix of stoichiometric coefficients of intermediates only. Number of intermediates in chemical system. 

IF ER=2 THEN LPRINT "RANK OF MATRIX IS LESS THAN NUMBER OF ATOMS"... 

The rank of matrix T can never be above N — m, where m is the number of chemical elements in the system. This holds due to the fact that there always exist m linearly dependent columns of matrix F set by eqn. (25)... 

The vector column for the matrix of stoichiometric numbers v (s-by-P) is called the route of a complex reaction. The rank of matrix Tint cannot be... 

Since with the fixed value of vector Pbr and the lack of constraint (62) the admissible region of solutions is a polyhedron, F reaches its minimum at one of its vertices. With the rank of matrix A equal to m-1 and n unknowns the reference solution contains no less than n- m-1) zero components, which equals the number of chords of the system of independent loops of the network graph. In this case the graph tree is a polyhedron vertex and the optimal variant should be among the set of trees of a redundant scheme. 

However, beeaiLse of Eq. (B.47) the elements of the nth column are identical to the ones forming column n+1 regardless of k such that all (n+1) x (n+1) subdeterminants of ip vanish on account of statement 4 of our list of elementary properties of determinants. This, in turn, proves that the rank of matrix ip is equal to the rank of matrix p. ... 

Rank of a matrix The rank of matrix A is the largest order of square array whose determinant is nonzero. Clearly, the rank of the matrix above cannot exceed the minimum of m and n. The definition of a determinant can be found in Amundson (1966). The matrix A may be regarded as a transformation of vectors in 91" into a range of vectors, denoted R(A), a subspace of 91 ". The rank of A may also be seen to be the dimension of R(A). When m = n, A is said to be a square matrix of order n. If its rank is less than n, it is said to be singular. Clearly, the determinant of the singular matrix is zero. 

In this case the rank of matrix Cst[3,5] is only two because the second element in the column matrix of eigenvalues ( EV[3,1]) is a lot smaller than the elements of the initial matrix and a lot smaller than the estimated accepted errors in expressing the standard concentrations. Consequently, even if a number of N = 5 standard solutions were used (with the considered concentrations), the concentrations of the three components in their mixture cannot be determined (irrespective of the wavelengths set chosen for measuring the absorbances), because the values of the concentrations of the standard solution have not been chosen properly. 

R, Ry, rank of matrix (general, of stoichiometrical coefficients and element-species), see App. I... 

Determination of the rank of matrix has been applied to several systems [31], and the results are briefly summarized in Table 3.2. Only the first example is discussed in detail for others we refer to literature [31]. Hogfeldt and Fredlund [39] studied the... 

The rank of matrix (9.3.39) is > // as rankA = H. Let it equal H thus 8j is linear combination of the rows of A . Then, given vector x obeying the solvability condition... 

S is the matrix (9.4.17). Recall the full row rank of matrix Aj, thus AjSA, is invertible. 

We can also compare the value (the same in both examples) with a critical value according to Sections 10.5 and 9.4 see (9.4.3). A gross error is detected if the value <2min exceeds the critical value %t (//) where H is the degree of freedom (here of redundancy) thus the rank of matrix A the event occurs... 

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