- #1
Dembadon
Gold Member
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I would like to check my reasoning for this problem to make sure I understand what an orthogonal matrix is.
Determine if the matrix is orthogonal. If orthogonal, find the inverse.
[tex]
\begin{pmatrix}
-1 & 2 & 2\\
2 & -1 & 2\\
2 & 2 & -1
\end{pmatrix}
[/tex]
If a matrix [itex]A[/itex] is orthogonal, then
[tex]
A^{-1} = A^T.
[/tex]
One of the conditions that must be met for a matrix to be orthogonal is that the length of the vectors spanning its column space must be 1, correct? So, if we let
[tex]
A=\begin{pmatrix}
-1 & 2 & 2\\
2 & -1 & 2\\
2 & 2 & -1
\end{pmatrix}=(\mathbf{a}_1,\mathbf{a}_2,\mathbf{a}_3),
[/tex]
then
[tex]
||\mathbf{a}_1||^2 \neq 1,
[/tex]
so a condition for orthogonality has been violated. Thus, [itex]A[/itex] is not orthogonal and there is no need to continue with the problem.
Another way to put it would be to say that the [itex]Col\ A[/itex] is not an orthonormal set, so [itex]A[/itex] is not orthogonal. Is this correct?
Homework Statement
Determine if the matrix is orthogonal. If orthogonal, find the inverse.
[tex]
\begin{pmatrix}
-1 & 2 & 2\\
2 & -1 & 2\\
2 & 2 & -1
\end{pmatrix}
[/tex]
Homework Equations
If a matrix [itex]A[/itex] is orthogonal, then
[tex]
A^{-1} = A^T.
[/tex]
The Attempt at a Solution
One of the conditions that must be met for a matrix to be orthogonal is that the length of the vectors spanning its column space must be 1, correct? So, if we let
[tex]
A=\begin{pmatrix}
-1 & 2 & 2\\
2 & -1 & 2\\
2 & 2 & -1
\end{pmatrix}=(\mathbf{a}_1,\mathbf{a}_2,\mathbf{a}_3),
[/tex]
then
[tex]
||\mathbf{a}_1||^2 \neq 1,
[/tex]
so a condition for orthogonality has been violated. Thus, [itex]A[/itex] is not orthogonal and there is no need to continue with the problem.
Another way to put it would be to say that the [itex]Col\ A[/itex] is not an orthonormal set, so [itex]A[/itex] is not orthogonal. Is this correct?