Orthogonal Matrix problems

**Runty** · March 9th 2010, 12:03 PM

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

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

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

ii) $A-B$
(I'm afraid I'm not sure about this one)

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

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

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?

**NonCommAlg** · March 9th 2010, 03:14 PM

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

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

ii) $A-B$
(I'm afraid I'm not sure about this one)

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

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

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

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

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

**Runty** · March 11th 2010, 05:10 AM

Originally Posted by NonCommAlg

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

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

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

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.

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