Homework Help

A square matrix A is called orthogonal if A^TA = I_n . Let v_1,v_2,.....,v_n be the...

user profile pic

hockeyfan54 | Student | Salutatorian

Posted April 10, 2013 at 10:17 PM via web

dislike 0 like

A square matrix A is called orthogonal if A^TA = I_n .

Let v_1,v_2,.....,v_n be the columns of an orthogonal matrix A . Show that the v_is are mutually perpendicular and unit vectors.

Tagged with linear algebra, math

1 Answer | Add Yours

user profile pic

pramodpandey | College Teacher | Valedictorian

Posted April 11, 2013 at 5:55 AM (Answer #1)

dislike 0 like

 `A=[[v_1,v_2,....,v_n]]`  and `A^TA=I_n`

The vectors `v_1,v_2,.....v_n` are perpendicular if

`v_i.v_j=0 if i!=j`

since matrix A is orthogonal ,there fore

`v^T_i.v_j=(i,j)^(th)` element of `I_n`

` =0`  if `i!=j`

Also

`v^T_i.v_i=(i,i)^(th)` element of `I_n`

`=1`


Hence proved

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes