Thread: 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.

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

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

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?

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