The Rank of a Matrix

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. 

In a casual way one could state that the rank of the matrix equals the number of different species that exist in the mixture. However, such a statement is not generally true and needs to be qualified in several ways  

A first question in model-free analysis is how many components are there in a system Or, in other words, what is the rank of the matrix Y In particular, what is the influence of noise Providing an answer to these questions is a first, extremely powerful result of SVD. 

There are many advantages in selecting only the significant ne eigenvectors and singular values for the representation of Y. In fact, from now on we only use this selection and introduce an appropriate nomenclature. 

Where U, V and S contain the significant parts of the total matrices U, S and V. The graphical representation is instructive  

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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... 

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). 

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. 

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... 

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. 

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... 

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. 

A matrix B is called singular if det B = 0, otherwise if det B 5 0 then B is non-singular. With this we can now introduce the concept of the rank of a matrix. The rank of an n x m matrix B, denoted rank(B), is defined to be the order of the largest non-singular square sub-matrix which can be formed by the selection of (possibly non-adjacent) rows and columns of B. For example. 

There is a direct relationship between the rank of a matrix and the linear independence of its components. That is, the rank of a matrix is equal to the... 

The rank of a matrix is the maximum number of linearly independent vectors (rows or columns) in an X/t matrix denoted as r X). Linearly dependent rows or columns reduce the rank of a matrix. 

The rank of a matrix A is the number of nonzero rows in the reduced row echelon form of A. 

J is equal to the rank of the matrix of the indices of the elements in the formulae for the constituents. The rank of a matrix is the order of the determinant of the highest order, which can be extracted from the matrix, which is not equal to zero. 

B.2.1.6 Rank The rank of a matrix A, rank(A), is a positive integer that counts the number of independent rows or columns in A. The row rank of A counts the number of independent rows in A, whereas the column rank of A counts the number of independent columns in A. A fundamental result of linear algebra is that the row and column rank of a matrix are always equal, and thus rank(A) = rank(A ). 

On the other hand, the rank of a matrix oi should accurately coincide with the rank of a matrix V, which determines the maximal number of independent combinations between component substances ... 

The rank of a matrix is defined as the order of the highest nonvanishing minor of that matrix (or, equivalently, the order of the highest square submatrix having a nonzero determinant). A square ra x n matrix is said to be of full rank if its rank is equal to n. The matrix is said to be rank deficient if the rank is less than n. 

We use the Rank matrix to calculate the rank of the Dimension matrix. We need the rank of the Dimension matrix to determine the number of independent solutions that exist for our system of linear equations. From linear algebra, the rank of a matrix is the number of linearly independent rows, or columns, of a matrix [24]. In other words the rank of a matrix is the number of independent equations in a system of linear equations. Thus the number of variables in a system of linear equations, that is, the number of columns in the Dimension matrix minus the rank of the Dimension matrix equals the number of selectable unknowns [23]. Mathematically... 

By elimination, the number of independent equations can be determined. Only when the number of variables equals the number of equations, while the equations are independent, can a non-singnlar solntion be fonnd. In that case the eliminated matrix is a full square matrix. In this context the concept rank is nseful. The rank of a matrix is the maximum number of independent rows (or, the maximnm nnmber of independent columns). A square matrix A( ,n) is non-singular only if its rank is equal to n. The rank can easily be found from the number of non-zero rows obtained by redncing the matrix to echelon form. 

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

The determinant, det( ) remains the proper measure of singularity. However, we might want some more information on just how singular a particular matrix A is if det(y4) = 0. The rank of a matrix A is the number of the linearly independent rows (or columns) of the matrix. Therefore, a nonsingular N x N matrix must be of rank N, and is said to be of full rank. The rank also may be defined as the dimension of the range of Matrix rank is... 

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