Rank of a matrix

The determination of the rank of a matrix is necessary to apply Brinkley and Jouguet s criteria. Let us recall that the rank of a matrix is the order of a non-zero determinant extracted from the matrix and of maximum order, or equivalently, the maximum number of independent columns or rows. Such definitions of rank are not useful in practice and methods based on transformations which do not modify rank are preferred. 

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Rank of a matrix The rank of a matrix is equal to the number of linearly independent rows or eolumns. The rank ean be found by determining the largest square... 

Dimensions and rank of a matrix are distinct concepts. A matrix can have relatively large dimensions say 100x50, but its rank can be small in comparison with its dimensions. This point can be made more clearly in geometrical terms. In a 100-dimensional row-space S ° , it is possible to represent the 50 columns of the matrix as 50 points, the coordinates of which are defined by the 100 elements in each of them. These 50 points form a pattern which we represent by P °. It is clear... 

Theorem Let A be an hermitian matrix. Then, the matrix D arising from the algorithm for calculating the rank of a matrix, i.e.,... 

Conversely, linear dependence occurs when some nonzero set of values for dj satisfies Equation (A.22). The rank of a matrix is defined as the number of linearly independent columns n). 

In order to analyze estimability utilizing such ideas, we first include some notions related to structure. Then we define the concepts of structural observability and the generic rank of a matrix. 

Within the Matlab s numerical precision X is singular, i.e. the two rows (and columns) are identical, and this represents the simplest form of linear dependence. In this context, it is convenient to introduce the rank of a matrix as the number of linearly independent rows (and columns). If the rank of a square matrix is less than its dimensions then the matrix is call rank-deficient and singular. In the latter example, rank(X)=l, and less than the dimensions of X. Thus, matrix inversion is impossible due to singularity, while, in the former example, matrix X must have had full rank. Matlab provides the function rank in order to test for the rank of a matrix. For more information on this topic see Chapter 2.2, Solving Systems of Linear Equations, the Matlab manuals or any textbook on linear algebra. 

The number of linearly independent columns (or rows) in a matrix is called the rank of that matrix. The rank can be seen as the dimension of the space that is spanned by the columns (rows). In the example of Figure 4-15, there are three vectors but they only span a 2-dimensional plane and thus the rank is only 2. The rank of a matrix is a veiy important property and we will study rank analysis and its interpretation in chemical terms in great detail in Chapter 5, Model-Free Analyses. 

The rank of a matrix Y is the number of linearly independent rows or columns in this matrix. The columns of Y are linearly dependent if one of the column vectors y j can be written as a linear combination of the other columns. The same holds for rows. 

Figure 5-3 displays two data matrices, used to demonstrate different ways of estimating the rank of a matrix. The top matrix has a noise level of 103 and the lower one of l.OlxlO1. The mean of all elements of Y is about 0.2 and the maximum is 2. Thus, the noise levels amount to some 0.5% and 50% of the mean and 0.05% and 5% of the maximal value of Y. 

In this section we review the known theorems that relate entanglement to the ranks of density matrices [52]. The rank of a matrix p, denoted as rank(p), is the maximal number of linearly independent row vectors (also column vectors) in the matrix p. Based on the ranks of reduced density matrices, one can derive necessary conditions for the separability of multiparticle arbitrary-dimensional mixed states, which are equivalent to sufficient conditions for entanglement [53]. For convenience, let us introduce the following definitions [54—56]. A pure state p of N particles Ai, A2,..., is called entangled when it cannot be written... 

The determination of the rank of a matrix is fairly simple and straight-forward. Unfortunately, the orthodox methods applied to a matrix such as A in Equation 2 give an answer which is exact mathematically but useless physically, namely that the rank of A is the number of radionuclides measured or the number of samples analyzed, whichever is less. This unfortunate result arises from presence of experimental imprecision in the elements of A. One must therefore rewrite Equation 2 in the form... 

Due to the special structure of MATLAB, readers should be familiar with the mathematical concepts pertaining to matrices, such as systems of linear equations, Gaussian elimination, size and rank of a matrix, matrix eigenvalues, basis change in n-dimensional space, matrix transpose, etc. For those who need a refresher on these topics there is a concise Appendix on linear algebra and matrices at the end of the book. 

If a set of independent vectors is multiplied by an orthogonal matrix, the resulting set is still independent. Thus, the ranks of A and 2 are the same. Consequently, the rank of a matrix is the number of non-zero singular values. 

The rank of a matrix is a mathematical concept that relates to the number of significant compounds in a dataset, in chemical terms to the number of compounds in a mixture. For example, if there are six compounds in a chromatogram, the rank of the data matrix from the chromatogram should ideally equal 6. However, life is never so simple. What happens is that noise distorts this ideal picture, so even though there may be only six compounds, either it may appear that the rank is 10 or more, or else the apparent rank might even be reduced if the distinction between the profiles for certain compounds are indistinguishable from the noise. If a 15 x 300 X matrix (which may correspond to 15 UV/vis spectra recorded at 1 nm intervals between 201 and 500 nm) has a rank of 6, the scores matrix T has six columns and the loadings matrix P has six rows. 

The concept of composition is an important one. There are many alternative ways of expressing the same idea, that of rank being popular also, which derives from matrices ideally the rank of a matrix equals the number of independent components or nonzero... 

Note how the rank of a (the rank of a matrix is given by the size of the largest nonzero determinant that can be formed from the matrix), the matrix composed of the first two columns, can at the most be 2 and that the rank of [a, b] is also 2 because the determinant of [a, b] is zero. To obtain a consistent set of equations, one of the three material balances must be eliminated, leaving two equations in two unknowns, P and W, that have a unique solution m = r — 2 and n = 2). The third equation is a redundant equation. It would probably be best to pick the two equations in which the coefficients were known with the greatest precision. 

The first row refers to the first reaction and the second row to the second reaction. The columns (species) are in the following order 1-CF2C12, 2-CF2ClH, 3-CF2H2, 4-H2, and 5-HCl. The rank of a matrix is the largest square submatrix obtained by deleting rows and columns, whose determinant is not zero. The rank equals the number of independent reactions. This is also equivalent to stating that there are reactions in the network that are linear combinations of the independent reactions. The rank of S above is 2, since the determinant of the first 2x2 submatrix is not zero (there are other 2x2 submatrices that are not zero as well but it is sufficient to have at least one that is not zero) ... 

The matrix (X - X) (X — X) is a real symmetric square matrix in which all elements are greater than or equal to zero. The rank of a matrix is the number of its non-zero eigenvalues. Imprecision due to an omnipresent error in all experimental measured data will give as result that, in practice, all eigenvalues will be different to zero and... 

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. 

Similarly, the number of linearly independent rows of A is called the row-rank of A. The row-rank of A is the column-rank of A. A fundamental theorem in matrix algebra states that the row-rank and the column-rank of a matrix are equal (and equal to the rank) [Schott 1997], Hence, it follows that the rank r(A) < min(/,/). The matrix A has full rank if and only if r(A) = mini/,/). Sometimes the term full column-rank is used. This means that r(A) = min(/,/) =. /, implying that J < I. The term full row-rank is defined analogously. 

There are different ways to implement a cross-validation scheme for two-way arrays [Eastment Krzanowski 1982, Wold 1978], An overview of methods to establish the pseudo-rank of a matrix is given by Faber [Faber et al. 1994],... 

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. 

The dimensional matrix is simply the matrix formed by tabulating the exponents of the fundamental dimensions M, L, and t, which appear in each of the variables involved. The rank of a matrix is the number of rows in the largest nonzero determinant which can be formed from it. An example of the evaluation of r and i, as well as the application of the Buckingham method, follows. 

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