# Thread: Orthogonal Matrix Preserve Length

1. ## Orthogonal Matrix Preserve Length

Ther question is:

Show that if Q is orthogonal, then Q preserves length. In other words, for all vectors x, the length of x equals the length of Qx.
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Ok, my understanding is that an orthogonal matrix is a square matrix where $P^T = P^{-1}$.

Also, the length of the column vectors or row vectors is equal to 1. At least, a theorem describes an n X n matrix P as being orthogonal if and only if the column vectors form an orthonormal set. Also, orthonormal seems to mean a set where the length equals 1 and the sets if dotted with each other will equal the zero vector.

I guess I am wanting confirmation. Because the columns and rows have length one, if we perform matrix multiplication with a vector x as described above, we effectively multiply that vector by one, thus preserving length. I am not quite sure how to answer the question other than saying that the length of the columns and rows are 1...and going from there. THANKS!

2. Notice the fact:
the square of the norm of x is equal to x*x,( * means transposition)