Mathematical Methods for Physics and Engineering : A Comprehensive Guide

8.17 QUADRATIC AND HERMITIAN FORMS as the necessary condition thatxmust satisfy. If (8.114) is satisfied for some eigenvectorxth ...

MATRICES AND VECTOR SPACES 8.17.2 Quadratic surfaces The results of the previous subsection may be turned round to state that th ...

8.18 SIMULTANEOUS LINEAR EQUATIONS where theAijandbihave known values. If all thebiare zero then the system of equations is call ...

MATRICES AND VECTOR SPACES If a vectorylies in the null space ofAthenAy= 0 ,whichwemaywriteas y 1 v 1 +y 2 v 2 +···+yNvN= 0. (8. ...

8.18 SIMULTANEOUS LINEAR EQUATIONS anyvector in the null space ofA(i.e.Ay= 0 )then A(x+y)=Ax+Ay=Ax+ 0 =b, and sox+yis also a sol ...

MATRICES AND VECTOR SPACES
Show that the set of simultaneous equations 2 x 1 +4x 2 +3x 3 =4, x 1 − 2 x 2 − 2 x 3 =0, (8.123) − ...

8.18 SIMULTANEOUS LINEAR EQUATIONS the nine elements of (8.126) to those of the 3×3 matrixA. This is done in the particular orde ...

MATRICES AND VECTOR SPACES This set of equations is also triangular, and we easily find the solution x 1 =2,x 2 =− 3 ,x 3 =4, wh ...

8.18 SIMULTANEOUS LINEAR EQUATIONS than just positive semi-definite) in order to perform the Cholesky decomposition (8.128). In ...

MATRICES AND VECTOR SPACES the unique solution. The proof given here appears to fail if any of the solutions xiis zero, but it c ...

8.18 SIMULTANEOUS LINEAR EQUATIONS (a) (b) Figure 8.1 The two possible cases whenAis of rank 2. In both cases all the normals li ...

MATRICES AND VECTOR SPACES the number of simultaneous equationsMis equal to the number of unknownsN. This technique is known ass ...

8.18 SIMULTANEOUS LINEAR EQUATIONS
Show that, fori=1, 2 ,...,p,Avi=siuiandA†ui=sivi,wherep=min(M, N). Post-multiplying both sid ...

MATRICES AND VECTOR SPACES non-zero singular valuessi,i=1, 2 ,...,r, then from the worked example above we have Avi=0 fori=r+1,r ...

8.18 SIMULTANEOUS LINEAR EQUATIONS We already know from the above discussion, however, that the non-zero eigenvalues of this mat ...

MATRICES AND VECTOR SPACES Using the unitarity of the matricesUandV, we find that Axˆ−b=USSU†b−b=U(SS−I)U†b. (8.142) The matrix ...

8.19 EXERCISES whereUandVare given by (8.139) and (8.140) respectively andSis obtained by taking the transpose ofSin (8.138) and ...

MATRICES AND VECTOR SPACES 8.3 Using the properties of determinants, solve with a minimum of calculation the following equations ...

8.19 EXERCISES (b) Without assuming thatBis orthogonal, prove thatAis singular. 8.9 Thecommutator[X,Y] of two matrices is define ...

MATRICES AND VECTOR SPACES (b) find an orthonormal basis, within a four-dimensional Euclidean space, for thesubspacespannedbythe ...