Proving transpose of orthogonal matrix orthogonal

[derryck1234]

Homework Statement

Show that if A is orthogonal, then A^T is orthogonal.

Homework Equations

AA^T = I

The Attempt at a Solution

I would go about this by letting A be an orthogonal matrix with a, b, c, d, e, f, g, h, i , j as its entries (I don't know how to draw that here)...but this would be a 3x3 matrix with entries a, b, c, d, e, f, g,h, i, j. I would then construct A^T, and then multiply the two matrices. I should find that the non-diagonal entries are zero, but how would I show that the diagonal entries are 1?

 

[Clever-Name]

Use the fact that the columns are all orthogonal unit vectors.

 

[derryck1234]

Ok. I understand how I would prove that A times the transpose of A has orthogonal columns...but how would I prove they are unit vectors?

 

[stringy]

What definition are you using for an orthogonal matrix? Note that orthogonal unit vectors for rows and columns is equivalent to $AA^T = A^TA=I$. The equivalence of these definitions is perhaps in your book or can certainly be found online.

I would use the second definition. A is orthogonal if and only if $AA^T=A^TA = I$. To show $A^T$ is orthogonal, make use of the fact that $(A^T)^T=A$.

 

[Clever-Name]

If you're assumed that A is already orthogonal then you don't need to prove that the columns are orthogonal unit vectors. That's the definition of an orthogonal matrix, thus already being in your assumption that A is orthogonal.