Operations on Matrices

Given an X m matrix, B, we can construct its transpose, B, by interchanging the rows and columns. Thus the ijth element of B becomes the jith element of B according to  

Taking the complex conjugate of every element of a matrix. A, yields the complex conjugate matrix, A that is, (A )/, = (A)/,. If all the elements of A are real, then A = A. 

The transpose of the complex conjugate matrix (sometimes termed the adjoint matrix), is written as A and defined such that  

Note these results are valid for any matrices A and B, for which multiplication is defined, 

The trace of a square matrix, A, of order n, denoted by trA, is defined as the sum of its diagonal elements  

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You can also create array formulas that return an array of different values in a selected range of cells. Formulas that operate on matrices, described in Chapter 9, are examples of formulas that return an array. 

The values comprising the array are called matrix elements. Mathematical operations on matrices have their own special rules. 

Operations on matrices containing real and complex elements. 

The controllability measures described in this section have a more general character. Measures applicable for square plants and decentralised control will be discussed in the next subchapter. The operations on matrices described below can be done easily with MATLAB, a specialised tool for analysing process control problems. 

The action of projection operators on matrices representing (Hermitian) operators clearly involves projection operators tujice because of the way in which an operator matrix transforms under a change of basis it is trivial to show that... 

Here are some functions for manipulating matrices. See help for details. Table 2.1 MATLAB built-in functions that are operable on matrices. 

MATLAB is designed to make operations on matrices as easy as possible. Most of the variables in MATLAB are considered as matrices. A scalar number is a 1x1 matrix and a vector is a Ixn (or nxl) matrix. Introducing a matrix is also done by an equality sign ... 

When we operate on an element by element basis, we need to add a period before the operator. Examples based on the two square matrices a and b ... 

The column-echelon form of G is obtained by performing a sequence of elementary column operations on this matrix. This means that we can find a sequence of elementary matrices EpEp i... Ei corresponding to the elementary column operations, such that... 

Thus the x matrix B is a function of B and therefore of r real numbers, which in our approach play the role of the parameters for A-representable 2-matrices within the limitations of the given one-particle basis set. Compare this with the = ( ) parameters of the FCl approach. Recall Kummer s basic theorem [1, Theorem 2.8, p. 56] that B could be a second-order RDM if and only if B ) is a positive operator on A-space. For 2, /i real and 2 > 0, we set... 

If a set of objects is such that operation on any of them by any element of a group produces a linear combination of the same set of elements, then the operations can be expressed in terms of a set of matrices that represent the... 

Exercise 2.30 Define an equivalence of matrices by Aj A2 if and only if there is a matrix B such that Ai = BA 2,B. Show that matrix multiplication is well defined on equivalence classes. Shoyv that trace and determinant are well defined on equivalence classes. Show that eigenvalues are well defined, but eigenvectors are not. Finally, show that given a vector space V, any linear operator on V corresponds to precisely one equivalence class of matrices. Exercise 2.31 Suppose V is a finite-dimensional vector space. 

Exercise 5.13 Can you use tensor products to construct a group operation on finite-sized square matrices of determinant one ... 

In this chapter we have shown that there are very many different sets of matrices which behave like the symmetry operations of a given point group. We have constructed these so-called representations by considering the action of the symmetry operations on a position vector or on any number of base vectors. Alternatively, we have found that we can find transformation operators Om which are homomorphic with the symmetry operations and that from these we can construct... 

We have shown that any set of n linearly independent functions that are transformed into linear combinations of one another by the symmetry operators of a group forms a basis for an n-dimensional representation of the group. The representation matrices consist of the coefficients describing the effect of the symmetry operations on the basis functions. 

We will not try to give a definite description or classification of mathematical objects here. This section should be regarded merely as a collection of useful facts and nomenclature. We will cover the most common terms regarding continuous spaces in general and vector spaces, operators and matrices. We will not touch upon spinors, nor on tensors. 

We can now list some of the most important properties of the various types of operators and matrices Any hermitean, antihermitean, unitary, or idempotent operator has a spectral resolution where the eigenvectors form an ON-basis, so that... 

Superficially, eqs. (3.1.11-12) seem to imply that diagonalization could be achieved in n(n - l)/2 steps. However, every time a rotation is used to set a specific At to zero, previously zeroed elements in the same row and column will be destroyed. This observation also hints at the proper selection of which element to eliminate in each step It is wasteful to spend operations on a rotation to eliminate a very small element which will anyway later get destroyed, so we should eliminate one of the largest elements. The best choice is usually the largest element however, for any but the smallest matrices, it takes more time to search for the largest element than to perform the rotation. Common practice is to simply loop over all elements in a predetermined order, but to slap the rotation for elements which fall below a certain threshold. This threshold is then successively lowered before each such sweep. Its value is based on the non-diagonality measure (3.1.6) (due to . Neumann) or some other convenient measure. 

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