# Orthogonality proof

• Mar 11th 2007, 07:17 PM
buckaroobill
Orthogonality proof
i have a linera algebra quiz on this proof tomorrow so if anyone could show me how the following is done, i would be really thankful...

An orthogonal matrix is one for which A^T = A^-1 (meaning A transpose = inverse of A).

Prove that if a matrix A is orthogonal, then any two distinct columns of A have dot product zero.

Is this true for the converse of this statement as well? Justify your answer.
• Mar 11th 2007, 09:05 PM
CaptainBlack
Quote:

Originally Posted by buckaroobill
i have a linera algebra quiz on this proof tomorrow so if anyone could show me how the following is done, i would be really thankful...

An orthogonal matrix is one for which A^T = A^-1 (meaning A transpose = inverse of A).

Prove that if a matrix A is orthogonal, then any two distinct columns of A have dot product zero.

Let A be orthogonal. and consider:

(A^t A)_{i,j}

this is the dot product of the i-th col and j-th col of A, hence as A^t=A^{-1}:

(A^t A)_{i,j} = 1, if i = j

(A^t A)_{i,j} = 0, if i != j

Quote:

Is this true for the converse of this statement as well? Justify your answer.
The converse is not true, Consider any orthogonal matrix A, then B=2A has the
given property but:

B^t B = 4 I.

RonL
• Mar 11th 2007, 09:08 PM
buckaroobill
Quote:

Originally Posted by CaptainBlack
Let A be orthogonal. and consider:

(A^t A)_{i,j}

this is the dot product of the i-th col and j-th col of A, hence as A^t=A^{-1}:

(A^t A)_{i,j} = 1, if i = j

(A^t A)_{i,j} = 0, if i != j

The converse is not true, Consider any orthogonal matrix A, then B=2A has the
given property but:

B^t B = 4 I.

RonL

Thanks! What do the underscores and the exclamation point preceding the equal sign mean, though?
• Mar 11th 2007, 11:59 PM
CaptainBlack
Quote:

Originally Posted by buckaroobill
Thanks! What do the underscores and the exclamation point preceding the equal sign mean, though?

A_{i,j} means A subscript i,j, it's LaTeX but our LaTeX is not working at present, but it is also a common ASCII math convention.

!= not equal to.

RonL