# Orthogonal Matrix problems

• Mar 9th 2010, 12:03 PM
Runty
Orthogonal Matrix problems
This problem I have partially solved, but I'm having issues concerning format and proper proofs. The question is listed below.

Let $\displaystyle A$ and $\displaystyle B$ be orthogonal $\displaystyle n\times n$ matrices. Determine which of the following are orthogonal (provide reasons for answers).

i) $\displaystyle A^{-1}$ (my answer is below, but I'm not sure if it is concrete enough)
Let $\displaystyle A=[a_{jk}]$. Since $\displaystyle A$ is orthogonal, then $\displaystyle A^{-1}=A^T=[a_{kj}]$. Therefore, $\displaystyle A^{-1}$ is orthogonal.

ii) $\displaystyle A-B$

If $\displaystyle AC$ is also an orthogonal $\displaystyle n\times n$ matrix, must $\displaystyle C$ be orthogonal?

I've inferred that $\displaystyle C$ is orthogonal, using the following three equivalent statements for some $\displaystyle n\times n$ matrix $\displaystyle Q$:
a. $\displaystyle Q$ is orthogonal.
b. $\displaystyle ||Qx||=||x||$ for every $\displaystyle x\in R^n$
c. $\displaystyle Qx\cdot Qy=x\cdot y$ for every $\displaystyle x,y\in R^n$

The issue I'm having is SHOWING how C is orthogonal without 'begging the question' (without using circular logic).

Can anyone help me with this?
• Mar 9th 2010, 03:14 PM
NonCommAlg
Quote:

Originally Posted by Runty
This problem I have partially solved, but I'm having issues concerning format and proper proofs. The question is listed below.

Let $\displaystyle A$ and $\displaystyle B$ be orthogonal $\displaystyle n\times n$ matrices. Determine which of the following are orthogonal (provide reasons for answers).

i) $\displaystyle A^{-1}$ (my answer is below, but I'm not sure if it is concrete enough)
Let $\displaystyle A=[a_{jk}]$. Since $\displaystyle A$ is orthogonal, then $\displaystyle A^{-1}=A^T=[a_{kj}]$. Therefore, $\displaystyle A^{-1}$ is orthogonal.

ii) $\displaystyle A-B$

If $\displaystyle AC$ is also an orthogonal $\displaystyle n\times n$ matrix, must $\displaystyle C$ be orthogonal?

I've inferred that $\displaystyle C$ is orthogonal, using the following three equivalent statements for some $\displaystyle n\times n$ matrix $\displaystyle Q$:
a. $\displaystyle Q$ is orthogonal.
b. $\displaystyle ||Qx||=||x||$ for every $\displaystyle x\in R^n$
c. $\displaystyle Qx\cdot Qy=x\cdot y$ for every $\displaystyle x,y\in R^n$

The issue I'm having is SHOWING how C is orthogonal without 'begging the question' (without using circular logic).

Can anyone help me with this?

i) $\displaystyle I=AA^T=(A^T)^T A^T.$ so $\displaystyle A^T=A^{-1}$ is orthogonal.

ii) not necessarly. for example we might have $\displaystyle A=B$.

iii) yes because we have $\displaystyle CC^T=A^TACC^TA^TA=A^TAC(AC)^TA=A^TIA=A^TA=I.$
• Mar 11th 2010, 05:10 AM
Runty
Quote:

Originally Posted by NonCommAlg
i) $\displaystyle I=AA^T=(A^T)^T A^T.$ so $\displaystyle A^T=A^{-1}$ is orthogonal.

ii) not necessarly. for example we might have $\displaystyle A=B$.

iii) yes because we have $\displaystyle CC^T=A^TACC^TA^TA=A^TAC(AC)^TA=A^TIA=A^TA=I.$

Thanks for those answers. Are you absolutely sure about the first one? It seems a little... incomplete to me. Not wrong, but a tad lacking in substance.