Let A 1 1 1 0 0 1 find two Givens rotators G1 G2 and B such

Let

A = (1    1

        1    0

        0    1)

find two Givens rotators G1, G2 and B such that

G2G1a1 = ( B

                    0

                    0 )

where a1 is the first column of A.

solve it by hand please!

Solution

APPLICATION OF ROTATION MATRIX TO ANY VECTOR WILL ALWAYS PRESERVES THE LENGTH .

WE HAVE |A1|= 2^0.5....SO ONE ROTATION OF THIS BY G1 & THEN G2 WILL STILL PRESEVE THE LENGTH AT 2^0.5 ONLY ...SO B=2^0.5 ....SINCE A1 IS [1,1,0] & FINAL PRODUCT IS [2^0.5,0,0] , WE NEED A PROJECTION MATRIX MULTIPLIED BY 2^0.5 FOR THE PURPOSE ..

THAT IS G2*G1=

2^0.5 , 0 , 0

0     , 0 , 0

0     , 0 , 0

SO WE HAVE TO FIND 2 ROTATION MATRICES WHOS COMPOSITE EFFECT IS TO GET THE ABOVE MATRIX

NOW THE OTHER PART , I HAVE TO STILL ANALYZE HOW TO DO BY HAND ..I SHALL COME BACK LATER..