Operator and Matrix Analysis 2.I INIRODUCTION In quantum mechanics and quantum statistics (quantum statistical mech- anics) as well as in certain other areas of physics, physical quantities are represented by linear operators on a vector (Hilbert) space. While this approach may seem far removed from an experimental process, it makes possible the development of these subject areas by use of rigorous mathe- matical procedures. Moreover, agreement between calculated results obtained by use of this formal (or abstract) approach with experimentally nieasured values has given credence to this formulation. Mathematical operations involving linear operators are often carried out by use of matrices. Hence a knowledge of vector spaces, linear operators, and matrix analysis is required in many areas of physics. We therefore begin this chapter with a brief discussion of vector spaces and linear operators; the remainder of the chapter is devoted to a development of matrix analysis. 42 Sec. 2.2 Rudiments of Vector Spaces 43 2.2 RUDIMENTS OF YECTOR SPACES 2.2.1 Definition of a Vector Spece A vector space (linear space or linear manifold) V" is a set of elements, V r Vrz, . , vn : {V}, called vectors for which the operations of addition and multiplication by a scalar defined below are valid. The term "vector," in this context, is used in an abstract mathematical sense; it is a generalization of the definition given in Chapter 1 to cases of arbitrary dimensions. In a strict mathematical sense, a space is a set of elements (vectors, points, func' tions, or any abstract quantities) for which certain defined mathematical operations are valid. We will call the set of elements vectors and require that they satisfy the following relations. A. Addirton of Yectors 1. For any two vectors yt, and tyr in Y* there exists the sum V, * ypinV, such that Vt*Vr:V**Vr 2. For vectors Vu V,, and yr in 2", there exists y, + (V r + 1r*) in Iz" such that Vr, * (Vt * V): (v' * ry) * v*. 3. There exists a unique vector 0 (zero or null vector) in Il such that vr*0:v for any y in V,. 4. For each vector ty in Vn, there exists a unique vector -V in Z, such that V * ?v):0. B. Mulrtphcation of Yectors by Scalars 5. For vectors Vrt nd ryr in Vn, there exists vectors u(V' * V), (a * f)V" and s,(fV) in Z" such that adV,*Y*):d.Yt*dtlr* (d 4- FW': e{t* Fv' o(fvt): (df)v, where d and f are two scalars (real or complex numbers). Opnuron aNp Mlrnx ANlr,ysls Csap.2 6. For the zero and unit vectors in V,,:1As following respective products exist: 0.V :0 and l.V : V. 2.2.2 Linear Dependence A set ofvectors [r4,] is said to be linearly dependent ifthere exists a correspond- ing set of scalars [a,], not all zero, such that utvt: o. (2.r) l=lE If p-fi':o implies that p,: 0 for all i, then the set of vectors [fJ is said to be linearly independent. Here {BJ is a set of scalars. 2.2.3 Dimensionality of a Vector Space A vector space is said to be z-dimensional if it contains precisely z linearly independent vectors. A vector space is called infinite-dimensional if there exists an arbitrarily large (but countable) number of linearly independent vectors in the space. If an arbitrary vector $ in V" can be represented as a linear combination of the vectors [y,] in V, and scalars [cJ u'v' (2.2) 6:2,= I then {r4,} is said to span the vector space V,. A linearly independent set of vectors [14,] that spans a vector space V, is called a basis for V,. For example, the i, j, k unit vectors described in Chapter I are the basis for the three-dimensional Cartesian space (a three- dimensional vector space). 2,2,4 Inner Product A Euclidean (Euclid about 300 B.c.) space E" is a vector space on which an inner (scalar) product is defined. The inner product of two vectors y and $ is denoted by (v,6) (2.3',) The following properties are valid for the inner product: Sec, 2.2 Rudiments of Yector SPaces 45 (v,6 * €): (v,0) * @,€) (v*6,O:(v,0+@,o and (V,V)>0 (unlesslr:0). r4 as The norm (length), ll ,lr I l, of a vector is definecl llyll : (v,v)'/'. (2'4) If the inner product of two vectors equals zero i' j : l'2' (vuvr): o for " ''tl I'*IV,+O it^ and Vr#O the vectors are said to form an orthogonal set. Ifthe norm within an orthogo- nal set is unity ll,r,ll: t the set is called orthonormal. 2.2,5 Hilbert (1862-1943) Space For a vector space v,to be complete, it is required that each cauchy sequence in V,, llV,-Vr|* 0 (for iandk * -), converges to a limit in Il. A complete and infinite-dimensional complex Euclidean space is called a Hilbert space. The notations used thus far in this section are those preferred by mathe- maticians. In physics (quantum mechanics), a Yector in Hilbert space is denoted by lV), and the inner product (V,6) is written as (t/l{) where lS)and (ylarecalledket and bra vectors, respectively. This latter notation is due to Dirac (1902-). 2.2.6 Linear Operators A linear operator on a vector space V, is a procedure for obtaining a unique vector, Qu in V, for each y, in Vo For example, 6': AVt (2.5a) where I is a linear operator. Using the Dirac notation, we write lD: tlv) (2.5b) Opsnaron enn MArnx ANlr,ysrs CH,lp.2 For linear operators A and A, it is required th4[ ,q(lvr) + ld>) : elD + el0) (A + B)lr): Alv) + Blv) GDID: A(Blv)) and AalV): q,AlV) where a is a scalar. The vectors ly) and lf) are two arbitrary vectors in the vector space. Exnuprr 2.1 A Quantum Mechanical lllustration of a Linear Operator Linear operators, in contrast to ordinary numbers and functions, do not always commute, that is, l,B is not always equal to BA. The difference AB - BA which is symbolically written as [A, B] is called the commutator of A and B. In quantum mechanics, linear operators play a central role, and it is understood that simultaneous specification of the physical quantities repre- sented by two noncommuting operators, lA, Bf * 0, cannot be made. Con- sider, for example, lx,p*j in the x-representation. Here x and p*, p,: -ihdl0x for i : 4/=l and h : hl2n, represent the position and x-compo- nent of the momentum of a particle, respectively. The value of the commuta- tor fx, p*l is obtained by operating on some function yr(.r); we obtain l*, p,lV@) : (xp* - p,x)V@) :-ih{.x-ry\ : ihty(x) or lx, p,7 : ih. The matrix representation of a linear operator is consistent with the fact that matrix multiplication is not in general commutative, as can be seen from the definition in Eq. (2.17). 2.3 MATRIX ANALYSIS AND NOTATIONS The word "matrix" was introduced in 1850 by Sylvester, and matrix theory was developed by Hamilton (1805-1865) and Cayley (1821-1895) in the study of simultaneous linear equations. Sec. 2.3 Malrix Analysis and Notations Matrices were rarely used by physicists prior to 1925. Today, they are used in most areas of physics. A matrix is a rectangular array of quantities, Qlo 4zo (2.6) where the at j are called elements (of the ith row and 7th column); they may be real (or complex) numbers or functions. The matrix A has m rows and n columns and is called a matrix of order m x n (m by n).lf m : n, the matrix is called a square matrix. The main diagonal of a square matrix consists of the elements cttr,ctzz,,,. rann. The row matrix A : (arr arz au) (2.7) is called a row vector, and the column matrix ^1',:,:) (2.8) is called a column vector. Two matrices of the same order (r4 and,B) are said to be equal if and only if a,t: b,, for all i and j. For example, ^:n and r:(rr,). (2.s) If a,,:0 for all i and j, then I is called a null matrix. For example, t0 0 0\ d:lo o ol. (2.10) \t ool The multiplication of a matrix, A, by a scalar, k, is given by kA: Ak (2.11) Opeuron eNo Mltntx AN,u,Ysts Ctrrp,2 where the elements of kA are ka,, for all i andT' For example, zf : e.t2) "\l '\t/ - \6f i\.zl 2.4 MATRIX OPERATIONS In this section, we discuss the important operations for matrices' 2.4,1 Addition (Subtraction) The operation of addition (or subtraction) for two n X zl matrices is defined as C:ATB (2.13) where c,, : a,t * b,, for all i and i. For example, :(1-l Q'\4, (i;l) ?) .(? ?-:) The following laws are also valid for addition of matrices of the same order: A+B:B*A (commutative) (2.15) (A+B)+c:A+(B+C) (associative) (2.16) 2.4.2 Multiplication For two matrices A and B,two kinds of products are defined; they are called the matrix product, AB, and direct product, A @ B. The matrix product c : AB is obtained by use of the following defini tion: (2.17) ,rr: ,f, o*b' where the orders of A,B,and Care r x s, s x nt, and n x respectively' '|1, Note that the matrix product is defined for conformable matrices only.