## Reduction of matrices to canonical form under generalized Lorentzian transformations |

### From inside the book

Results 1-3 of 12

Page 20

T.k-1 (0),T_ wx 'o u =w I[(A+in I) u J = + [ (A - l \x I) w J Ju ^ (k-1) "W . (0) (k-1) = +

W o u = *U © W

2. 5) (0) (0) express ion*non-zero. We note that u + cw is not in N((A - i n I) k_1 ) ...

T.k-1 (0),T_ wx 'o u =w I[(A+in I) u J = + [ (A - l \x I) w J Ju ^ (k-1) "W . (0) (k-1) = +

W o u = *U © W

**hence**, by an appropriate choice of c as 1 or i , we can make the (2. 5) (0) (0) express ion*non-zero. We note that u + cw is not in N((A - i n I) k_1 ) ...

Page 21

As in the proof of Lemma 9, it is simple to show that modifications of the type we

have just performed do not destroy existing orthogonality relations;

finite number of steps, we obtain a vector of the desired kind. The final two

lemmas ...

As in the proof of Lemma 9, it is simple to show that modifications of the type we

have just performed do not destroy existing orthogonality relations;

**hence**, in afinite number of steps, we obtain a vector of the desired kind. The final two

lemmas ...

Page 43

3 ] = (-l))-12x°-l>.y<k-"M-l)lc-Wk-1»f y(0»*0, J = l,2,...,k, v 0 v k+j k+j

form a standard set. Clearly x(J+1) =(A^I)x(j) y(j+1) = (A+nI)y(j) j = 0,l,...,k-l, or

equivalently Ax()» =x<)+1» ♢ *y<j)=y()+1)-,y(» 1-W.....W. Thus we obtain the

results ...

3 ] = (-l))-12x°-l>.y<k-"M-l)lc-Wk-1»f y(0»*0, J = l,2,...,k, v 0 v k+j k+j

**Hence**theform a standard set. Clearly x(J+1) =(A^I)x(j) y(j+1) = (A+nI)y(j) j = 0,l,...,k-l, or

equivalently Ax()» =x<)+1» ♢ *y<j)=y()+1)-,y(» 1-W.....W. Thus we obtain the

results ...

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Common terms and phrases

blocks have 1/2 canonical blocks canonical form canonical matrices center block characteristic polynomial Corollary corresponding factor define vj diag diagonal blocks direct sum DOCTOR OF PHILOSOPHY elementary divisors exists a standard exists a vector finite number form a standard hence imply index 4k invariant subspace iy(j J-isotropic subspace J-Lorentzian matrix J-orthogonal complement J-orthonormal basis J-skew matrix J-symmetric and J-skew Lemma 9 MacDuffee MATRICES TO CANONICAL minimum polynomial nilpotent J-skew matrices non-principal diagonal non-singular matrix non-zero J-value obtain a vector odd order orthogonal matrix permutation matrix positive definite principal diagonal proof of Theorem pure imaginary characteristic real vectors reduction of J-symmetric secondary diagonal set of index set of real standard set symmetric and skew T-symmetric matrix Theorem 13 thesis University of Wisconsin vector in N((A vectors of index vk+1 vk+2 yx Jy zeros elsewhere