The determinant of a matrix

In this formula, the choice of the row index i is indifferent and A, is the i/ th minor, i.e., the determinant of the matrix obtained by deleting the ith row and the /th column of the matrix A. The recurrence relation requires a starting condition the determinant of a scalar a, i.e., of a matrix of dimension 1 x 1, is a. The product (— 1),+J A0 is the cofactor of rank ij. The development can be equally made column-wise, i.e., 

If det A =0, the column-vectors A are not linearly independent (nor are the column-rows) and the matrix is singular. At least one edge-vector of the hyper-prism made of the column-vectors is in the subspace of the remaining edge-vectors the volume of the hyper-prism vanishes and det A = 0. 

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Determinants have many useful and interesting properties. The determinant of a matrix is ero if any two of its rows or columns are identical. The sign of the determinant is reversed )y exchanging any pair of rows or any pair of columns. If all elements of a row (or column) ire multiplied by the same number, then the value of the determinant is multiplied by that lumber. The value of a determinant is unaffected if equal multiples of the values in any row or column) are added to another row (or column). 

The anti symmetrized orbital produet A (l)i(l)2Cl)3 is represented by the short hand (1>1(1>2(1>3 I and is referred to as a Slater determinant. The origin of this notation ean be made elear by noting that (1/VN ) times the determinant of a matrix whose rows are labeled by the index i of the spin-orbital (jii and whose eolumns are labeled by the index j of the eleetron at rj is equal to the above funetion A (l)i(l)2Cl)3 = (1/V3 ) det(( )i (rj)). The general strueture of sueh Slater determinants is illustrated below ... 

Til explain later how we characterize such surfaces you might have noticed that (1.45) can be written as the determinant of a matrix H called the Hessian... 

The value of a determinant is unchanged if the rows are written as columns. Thus, the determinants of a matrix A and its transpose matrix are equal. 

For small determinantal wavefunctions these statements are easily verified by explicit expansion the general proof rests on the fact that the determinant of a matrix product is equal to the product of the determinants of the matrices. 

Note that = det is the determinant of the 3N x 3N transformation matrix 8R /8g , which gives the Jacobian for the transformation from generalized to Cartesian coordinates. This follows from the fact that the right-hand side (RHS) of Eq. (2.16) for g p is a matrix product of this transformation matrix with its transpose, and that the determinant of a matrix product is a product of determinants. By similar reasoning, we find that... 

Problem 8-9. If a two-particle wave function has the form of a product, f2) =

[Pg.73]

The definition of the determinant of a linear operator is analogous to the definition of the trace. We start with the determinant of a matrix, which should be familiar from a Unear or abstract algebra textbook such as Artin [Ar, Section 1.3], It is a fact of linear algebra that det(AB) = (det A)(det B) for any two square matrices A and B of the same size. Hence for any matrices A and A related by Equation 2.5, we have... 

As before, this calculation allows us to use the determinant of a matrix to define the determinant of a linear operator. 

So, for example, we can use the matrix of Formrrla 2.3 to see that the determinant of our favorite rotation is (0)(0) — (1)(—1) = 1. Note that we could just as well have used the matrix of Formula 2.4 to calculate the same answer (0) (0) — (2)(— ) = 1. No one familiar with the geometric interpretation of the determinant will be surprised by this result the determinant of a matrix with real entries is always the signed volume of the image of the unit square (or cube, or higher-dimensional cube), with a negative sign if the linear transformation changes the orientation. For more on this topic, see Lax [La, Chapter 5]. 

The determinant of a matrix equals the product of its characteristic roots, so the log determinant equals the sum of the logs of the roots. The characteristic roots of the matrix above remain to be detennined. As shown in the exercise, T-1 of the T roots equal 1. Therefore, the logs of these roots are zero, so the log-detenninant equals the log of the remaining root. It remains only to find the other characteristic root. Premultiply the result (o 2/a 2)ii c = (A-1 )c by i to obtain (a 2/a 2)i ii c = (A-l)i c. 

Now, we note that the determinants of matrices have a property that det(AB) = det(BA) = (detA)(detB), and hence, the determinant of a matrix remains intact by a unitary transformation of this matrix. Using this fact, one may show that the eigenphase sum and the background eigenphase sum may be related in a simple way to the determinants of the S and Sb matrices as... 

In his classic paper on electric networks, G. Kirchhoff[38] (1847) implicitly established the celebrated Matrix-Tree-Theorem which, in modern terminology, expresses the complexity (i.e., the number of spanning trees) of any finite graph G as the determinant of a matrix which can easily be obtained from the adjacency matrix of G. Simple proofs were given by R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte [39] (1940), H. Trent [40] (1954), and H. Hutschenreuther [41] (1967) (for relations between the complexity and the spectrum of a graph see Ref. [36] pp 38, 39, 49, 50). 

The isotropic-to-nematic transition is defined by the characteristic equation Det M = 0 (where Det represents the determinant of a matrix). If the Van der Waals interactions were turned off (W0 = 0) so that only nematic interactions are left, then M would be the denominator of X so that X would blow up for this condition (Det M = 0). Above certain critical values of Wj s the blend forms the nematic phase. As in the case of purely flexible mixtures, the spinodal condition is ... 

Some molecular descriptors, called - determinant-based descriptors, are calculated as the determinant of a - matrix representation of a molecular structure. Moreover, permanents, short- and long-hafnians, calculated on the topological - distance matrix D, were used as graph invariants by Schultz and called per(D) index, shaf(D) index, lhaf(D) index [Schultz et al, 1992 Schultz and Schultz, 1992]. 

Descriptors obtained by the calculation of the -> determinant of a matrix representing a - molecular graph. Molecular descriptors similar to the determinant-based descriptors could be obtained applying -> permanent and - hafnian to any matrix representing a molecular graph, such as -> per(D) index, - shaf(D) index, and -> Ihaf(D) index [Schultz et ai, 1992 Schultz and Schultz, 1992]. 

This important concept is defined for quadratic matrices, i = j. The determinant of a matrix A is denoted... 

For the determinant of a matrix A and its inverse A the following realtion applies deiA" = 1/detA... 

The determinant. This is an easy one, because the determinant of a matrix is a scalar (a single number) so you need not use the Ctrl + Shift + Enter trick. Just go to any empty cell, deposit the instruction = MDETERM(A1 D4) and (yes) press Enter. The answer, -196, will appear. 

The determinant of a matrix, denoted x, is a scalar measure of the overall volume of a matrix. For variance-covariance matrices, the determinant sometimes expresses the generalized variance of a matrix. In the simplest case, a 2 x 2 matrix, the determinant is calculated by... 

Determinant The determinant of a matrix is a function that assigns a number to a matrix. For example, the determinant of the n X n matrix B is represented by... 

The determinant of a matrix is equal to the determinant of the transpose of that matrix. 

The determinant of a matrix exists only for a square matrix and is a number. The determinant of a matrix is written as A. For a 2 x 2 matrix it is simple to define ... 

We assume that you have had some exposure to matrices so that the standard matrix operations are familiar to you. All you need to remember is how the inverse of a matrix, the determinant of a matrix, and the transpose of a matrix are calculated and how to add, subtract, and multiply matrices. 

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