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.



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Posted on (Answer #1)

 `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`


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


Hence proved

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