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Thread: Orthogonal Matrix problems

  1. #1
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    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) $\displaystyle A-B$
    (I'm afraid I'm not sure about this one)

    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?
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  2. #2
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    Quote Originally Posted by Runty View Post
    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) $\displaystyle A-B$
    (I'm afraid I'm not sure about this one)

    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 $\displaystyle CC^T=A^TACC^TA^TA=A^TAC(AC)^TA=A^TIA=A^TA=I.$
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  3. #3
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    Quote Originally Posted by NonCommAlg View Post
    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.
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