- #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.
<br /> \begin{pmatrix}<br /> -1 & 2 & 2\\<br /> 2 & -1 & 2\\<br /> 2 & 2 & -1<br /> \end{pmatrix}<br />
If a matrix A is orthogonal, then
<br /> A^{-1} = A^T.<br />
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
<br /> A=\begin{pmatrix}<br /> -1 & 2 & 2\\<br /> 2 & -1 & 2\\<br /> 2 & 2 & -1<br /> \end{pmatrix}=(\mathbf{a}_1,\mathbf{a}_2,\mathbf{a}_3),<br />
then
<br /> ||\mathbf{a}_1||^2 \neq 1,<br />
so a condition for orthogonality has been violated. Thus, A is not orthogonal and there is no need to continue with the problem.
Another way to put it would be to say that the Col\ A is not an orthonormal set, so A is not orthogonal. Is this correct?
Homework Statement
Determine if the matrix is orthogonal. If orthogonal, find the inverse.
<br /> \begin{pmatrix}<br /> -1 & 2 & 2\\<br /> 2 & -1 & 2\\<br /> 2 & 2 & -1<br /> \end{pmatrix}<br />
Homework Equations
If a matrix A is orthogonal, then
<br /> A^{-1} = A^T.<br />
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
<br /> A=\begin{pmatrix}<br /> -1 & 2 & 2\\<br /> 2 & -1 & 2\\<br /> 2 & 2 & -1<br /> \end{pmatrix}=(\mathbf{a}_1,\mathbf{a}_2,\mathbf{a}_3),<br />
then
<br /> ||\mathbf{a}_1||^2 \neq 1,<br />
so a condition for orthogonality has been violated. Thus, A is not orthogonal and there is no need to continue with the problem.
Another way to put it would be to say that the Col\ A is not an orthonormal set, so A is not orthogonal. Is this correct?
