# orthogonal matrix

• Mar 17th 2008, 01:19 AM
b00yeah05
orthogonal matrix
Show that matrix A is orthogonal...

I know that to be orthogonal the dot product must equal to 0.
but how do i go on about showing that this matrix is orthogonal??

http://img252.imageshack.us/img252/4...rixaal4.th.jpg
• Mar 17th 2008, 01:50 AM
Moo
Hello,

A matrix is orthogonal if \$\displaystyle MM^t = M^t M = Id\$

If \$\displaystyle M=(a_{ij})\$, \$\displaystyle M^t = (a_{ji})\$
• Mar 17th 2008, 01:52 AM
earboth
Quote:

Originally Posted by b00yeah05
Show that matrix A is orthogonal...

I know that to be orthogonal the dot product must equal to 0.
but how do i go on about showing that this matrix is orthogonal??

http://img252.imageshack.us/img252/4...rixaal4.th.jpg

Consider the rows of the matrix (or the columns - it doesn't matter) as vectors.

If each of these 3 vectors have the length 1 and form an orthonormal basis then the matrix is orthogonal.
• Mar 17th 2008, 02:21 AM
b00yeah05
from my definition it says that it is orthonormal if...

||vi|| = 1 or vi.vi = 1

so to work out the length i get the sqrt of the squred values..but what do you mean by orthonormal basis in your comment there? im guessing that it is important as its in bold..
• Mar 17th 2008, 03:09 AM
mr fantastic
Quote:

Originally Posted by b00yeah05
from my definition it says that it is orthonormal if...

||vi|| = 1 or vi.vi = 1

so to work out the length i get the sqrt of the squred values..but what do you mean by orthonormal basis in your comment there? im guessing that it is important as its in bold..

The v's must be mutually perpendicular .... v1 . v2 = 0, v1 . v3 = 0, v2 . v3 = 0