Orthogonal Matrices - Linear Algebra

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Question

Determine if the following matrix is orthogonal or not.

$A=\begin{bmatrix} -1& 0\ 0& 1 \end{bmatrix}$

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Answer

To determine if a matrix is orthogonal, we need to multiply the matrix by it's transpose, and see if we get the identity matrix.

$A=\begin{bmatrix} -1& 0\ 0& 1 \end{bmatrix}$ ,

Since we get the identity matrix, then we know that is an orthogonal matrix.

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Question

Which of the matrices is orthogonal?

$\begin{pmatrix} 1&0 \ 0&1 \end{pmatrix}$ , $\begin{pmatrix} 1&1 \ 1&1 \end{pmatrix}$ , $\begin{pmatrix} 1&1 \ 0&0 \end{pmatrix}$ , $\begin{pmatrix} 0&0 \ 1&1 \end{pmatrix}$

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Answer

An x matrix is defined to be orthogonal if

where is the x identity matrix.

We see that

$\begin{pmatrix} 1&0 \ 0&1 \end{pmatrix}$ $\begin{pmatrix} 1&0 \ 0&1 \end{pmatrix}=\begin{pmatrix} 11+00&10+01 \ 01+10&00+11 \end{pmatrix}$

$=\begin{pmatrix} 1&0 \ 0&1 \end{pmatrix}$

And so

$\begin{pmatrix} 1&0 \ 0&1 \end{pmatrix}$ is orthogonal.

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Question

Which of the matrices is orthogonal?

$\begin{pmatrix} 0&0 \ 0&0 \end{pmatrix}$ , $\begin{pmatrix} 1&0 \ 0&-1 \end{pmatrix}$ , $\begin{pmatrix} 1&1 \ 0&0 \end{pmatrix}$ , $\begin{pmatrix} 0&0 \ 1&1 \end{pmatrix}$

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Answer

An x matrix is defined to be orthogonal if

where is the x identity matrix.

We see that

$\begin{pmatrix} 1&0 \ 0&-1 \end{pmatrix}$ $\begin{pmatrix} 1&0 \ 0&-1 \end{pmatrix}=\begin{pmatrix} 11+00&10+0-1 \ 01+(-1)0&00+(-1)(-1) \end{pmatrix}$

$=\begin{pmatrix} 1&0 \ 0&1 \end{pmatrix}$

And so

$\begin{pmatrix} 1&0 \ 0&-1 \end{pmatrix}$ is orthogonal.

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Question

$\begin{align}&$\text{Conclude whether or not the matrix }$A=\begin{bmatrix}-0.60587&0.79556\0.79556&0.60587\end{bmatrix}\&$\text{is orthogonal.}$\end{align}$

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Answer

$\begin{align}&$\text{There are multiple ways to determine whether or not a matrix is orthogonal.}$\&$\text{For instance, a matrix is orthogonal if its transpose equals its $inverse:}$\&A^{-1}$$=A^{T}$\&$\text{Another method is to see if the product of it and its tranpose is the identity $matrix:}$\&A^{T}$$A=AA^{T}$=I\&$\text{If so, it is orthgonal.}$\&$\text{Finally, if the matrix's determinant is }$\pm1\&$\text{then it is orthgonal. Using this last method:}$\&det\begin{bmatrix}-0.60587&0.79556\0.79556&0.60587\end{bmatrix}= \&$\text{The matrix is orthogonal.}$\end{align}$

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Question

$\begin{align}&$\text{Conclude whether or not the matrix }$A=\begin{bmatrix}0.68567&0.12975&-0.71626\0.14807&0.93855&0.31176\0.71269&-0.31982&0.62433\end{bmatrix}\&$\text{is orthogonal.}$\end{align}$

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Answer

$\begin{align}&$\text{There are multiple ways to determine whether or not a matrix is orthogonal.}$\&$\text{For instance, a matrix is orthogonal if its transpose equals its $inverse:}$\&A^{-1}$$=A^{T}$\&$\text{Another method is to see if the product of it and its tranpose is the identity $matrix:}$\&A^{T}$$A=AA^{T}$=I\&$\text{If so, it is orthgonal.}$\&$\text{Finally, if the matrix's determinant is }$\pm1\&$\text{then it is orthgonal. Using this last method:}$\&det\begin{bmatrix}0.68567&0.12975&-0.71626\0.14807&0.93855&0.31176\0.71269&-0.31982&0.62433\end{bmatrix}=1\&$\text{The matrix is orthogonal.}$\end{align}$

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Question

$\begin{align}&$\text{Is }$A=\begin{bmatrix}-0.85616&0.46933\0.46933&0.96236\end{bmatrix}\&$\text{orthogonal?}$\end{align}$

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Answer

$\begin{align}&$\text{There are multiple ways to determine whether or not a matrix is orthogonal.}$\&$\text{For instance, a matrix is orthogonal if its transpose equals its $inverse:}$\&A^{-1}$$=A^{T}$\&$\text{Another method is to see if the product of it and its tranpose is the identity $matrix:}$\&A^{T}$$A=AA^{T}$=I\&$\text{If so, it is orthgonal.}$\&$\text{Finally, if the matrix's determinant is }$\pm1\&$\text{then it is orthgonal. Using this last method:}$\&det\begin{bmatrix}-0.85616&0.46933\0.46933&0.96236\end{bmatrix}=-1.0442\&$\text{The matrix is not orthogonal.}$\end{align}$

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Question

$\begin{align}&$\text{Conclude whether or not the matrix }$A=\begin{bmatrix}-0.57605&-0.81742\-0.81742&0.57605\end{bmatrix}\&$\text{is orthogonal.}$\end{align}$

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Answer

$\begin{align}&$\text{There are multiple ways to determine whether or not a matrix is orthogonal.}$\&$\text{For instance, a matrix is orthogonal if its transpose equals its $inverse:}$\&A^{-1}$$=A^{T}$\&$\text{Another method is to see if the product of it and its tranpose is the identity $matrix:}$\&A^{T}$$A=AA^{T}$=I\&$\text{If so, it is orthgonal.}$\&$\text{Finally, if the matrix's determinant is }$\pm1\&$\text{then it is orthgonal. Using this last method:}$\&det\begin{bmatrix}-0.57605&-0.81742\-0.81742&0.57605\end{bmatrix}=-1\&$\text{The matrix is orthogonal.}$\end{align}$

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Question

$\begin{align}&$\text{Is the matrix }$A=\begin{bmatrix}-0.44125&-0.89739\-0.89739&0.44125\end{bmatrix}\&$\text{orthogonal?}$\end{align}$

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Answer

$\begin{align}&$\text{There are multiple ways to determine whether or not a matrix is orthogonal.}$\&$\text{For instance, a matrix is orthogonal if its transpose equals its $inverse:}$\&A^{-1}$$=A^{T}$\&$\text{Another method is to see if the product of it and its tranpose is the identity $matrix:}$\&A^{T}$$A=AA^{T}$=I\&$\text{If so, it is orthgonal.}$\&$\text{Finally, if the matrix's determinant is }$\pm1\&$\text{then it is orthgonal. Using this last method:}$\&det\begin{bmatrix}-0.44125&-0.89739\-0.89739&0.44125\end{bmatrix}=-1\&$\text{The matrix is orthogonal.}$\end{align}$

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Question

$\begin{align}&$\text{Is the matrix }$A=\begin{bmatrix}-0.62126&-0.0069882&0.77059\-0.86406&-0.14727&-0.61555\0.011558&-1.0415&0.043005\end{bmatrix}\&$\text{orthogonal?}$\end{align}$

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Answer

$\begin{align}&$\text{There are multiple ways to determine whether or not a matrix is orthogonal.}$\&$\text{For instance, a matrix is orthogonal if its transpose equals its $inverse:}$\&A^{-1}$$=A^{T}$\&$\text{Another method is to see if the product of it and its tranpose is the identity $matrix:}$\&A^{T}$$A=AA^{T}$=I\&$\text{If so, it is orthgonal.}$\&$\text{Finally, if the matrix's determinant is }$\pm1\&$\text{then it is orthgonal. Using this last method:}$\&det\begin{bmatrix}-0.62126&-0.0069882&0.77059\-0.86406&-0.14727&-0.61555\0.011558&-1.0415&0.043005\end{bmatrix}=1.0968\&$\text{The matrix is not orthogonal.}$\end{align}$

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Question

$\begin{align}&$\text{Is the matrix }$A=\begin{bmatrix}0.012505&-0.9358&-0.11296\-0.84627&0.011244&0.49169\-0.3916&0.21287&-0.83217\end{bmatrix}\&$\text{orthogonal?}$\end{align}$

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Answer

$\begin{align}&$\text{There are multiple ways to determine whether or not a matrix is orthogonal.}$\&$\text{For instance, a matrix is orthogonal if its transpose equals its $inverse:}$\&A^{-1}$$=A^{T}$\&$\text{Another method is to see if the product of it and its tranpose is the identity $matrix:}$\&A^{T}$$A=AA^{T}$=I\&$\text{If so, it is orthgonal.}$\&$\text{Finally, if the matrix's determinant is }$\pm1\&$\text{then it is orthgonal. Using this last method:}$\&det\begin{bmatrix}0.012505&-0.9358&-0.11296\-0.84627&0.011244&0.49169\-0.3916&0.21287&-0.83217\end{bmatrix}=0.85765\&$\text{The matrix is not orthogonal.}$\end{align}$

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Question

$\begin{align}&$\text{Is }$A=\begin{bmatrix}-0.55978&-0.80992&0.17514\0.55713&-0.52432&-0.64396\0.61339&-0.2629&0.74474\end{bmatrix}\&$\text{orthogonal?}$\end{align}$

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Answer

$\begin{align}&$\text{There are multiple ways to determine whether or not a matrix is orthogonal.}$\&$\text{For instance, a matrix is orthogonal if its transpose equals its $inverse:}$\&A^{-1}$$=A^{T}$\&$\text{Another method is to see if the product of it and its tranpose is the identity $matrix:}$\&A^{T}$$A=AA^{T}$=I\&$\text{If so, it is orthgonal.}$\&$\text{Finally, if the matrix's determinant is }$\pm1\&$\text{then it is orthgonal. Using this last method:}$\&det\begin{bmatrix}-0.55978&-0.80992&0.17514\0.55713&-0.52432&-0.64396\0.61339&-0.2629&0.74474\end{bmatrix}=1\&$\text{The matrix is orthogonal.}$\end{align}$

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Question

$\begin{align}&$\text{Determine if the matrix }$A=\begin{bmatrix}-1.2228&-0.0071287&-1.0176\-1.6301&-1.11&-0.23163\-0.26127&-0.76287&-0.30747\end{bmatrix}\&$\text{is orthogonal.}$\end{align}$

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Answer

$\begin{align}&$\text{There are multiple ways to determine whether or not a matrix is orthogonal.}$\&$\text{For instance, a matrix is orthogonal if its transpose equals its $inverse:}$\&A^{-1}$$=A^{T}$\&$\text{Another method is to see if the product of it and its tranpose is the identity $matrix:}$\&A^{T}$$A=AA^{T}$=I\&$\text{If so, it is orthgonal.}$\&$\text{Finally, if the matrix's determinant is }$\pm1\&$\text{then it is orthgonal. Using this last method:}$\&det\begin{bmatrix}-1.2228&-0.0071287&-1.0176\-1.6301&-1.11&-0.23163\-0.26127&-0.76287&-0.30747\end{bmatrix}=-1.1685\&$\text{The matrix is not orthogonal.}$\end{align}$

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Question

$\begin{align}&$\text{Is }$A=\begin{bmatrix}-0.51053&0.15898&-0.84503\-0.61849&-0.75062&0.23245\-0.59735&0.64132&0.48155\end{bmatrix}\&$\text{orthogonal?}$\end{align}$

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Answer

$\begin{align}&$\text{There are multiple ways to determine whether or not a matrix is orthogonal.}$\&$\text{For instance, a matrix is orthogonal if its transpose equals its $inverse:}$\&A^{-1}$$=A^{T}$\&$\text{Another method is to see if the product of it and its tranpose is the identity $matrix:}$\&A^{T}$$A=AA^{T}$=I\&$\text{If so, it is orthgonal.}$\&$\text{Finally, if the matrix's determinant is }$\pm1\&$\text{then it is orthgonal. Using this last method:}$\&det\begin{bmatrix}-0.51053&0.15898&-0.84503\-0.61849&-0.75062&0.23245\-0.59735&0.64132&0.48155\end{bmatrix}=1\&$\text{The matrix is orthogonal.}$\end{align}$

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Question

$\begin{align}&$\text{Determine if the matrix }$A=\begin{bmatrix}-0.56605&-0.7545&-0.33214\0.76145&-0.32415&-0.56136\0.31588&-0.57066&0.758\end{bmatrix}\&$\text{is orthogonal.}$\end{align}$

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Answer

$\begin{align}&$\text{There are multiple ways to determine whether or not a matrix is orthogonal.}$\&$\text{For instance, a matrix is orthogonal if its transpose equals its $inverse:}$\&A^{-1}$$=A^{T}$\&$\text{Another method is to see if the product of it and its tranpose is the identity $matrix:}$\&A^{T}$$A=AA^{T}$=I\&$\text{If so, it is orthgonal.}$\&$\text{Finally, if the matrix's determinant is }$\pm1\&$\text{then it is orthgonal. Using this last method:}$\&det\begin{bmatrix}-0.56605&-0.7545&-0.33214\0.76145&-0.32415&-0.56136\0.31588&-0.57066&0.758\end{bmatrix}=1\&$\text{The matrix is orthogonal.}$\end{align}$

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Question

$\begin{align}&$\text{Is the matrix }$A=\begin{bmatrix}-0.58762&-0.34601&-1.5258\0.13555&-1.2568&-0.72109\-1.2767&-1.2221&-0.65535\end{bmatrix}\&$\text{orthogonal?}$\end{align}$

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Answer

$\begin{align}&$\text{There are multiple ways to determine whether or not a matrix is orthogonal.}$\&$\text{For instance, a matrix is orthogonal if its transpose equals its $inverse:}$\&A^{-1}$$=A^{T}$\&$\text{Another method is to see if the product of it and its tranpose is the identity $matrix:}$\&A^{T}$$A=AA^{T}$=I\&$\text{If so, it is orthgonal.}$\&$\text{Finally, if the matrix's determinant is }$\pm1\&$\text{then it is orthgonal. Using this last method:}$\&det\begin{bmatrix}-0.58762&-0.34601&-1.5258\0.13555&-1.2568&-0.72109\-1.2767&-1.2221&-0.65535\end{bmatrix}=2.3855\&$\text{The matrix is not orthogonal.}$\end{align}$

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Question

By definition, an orthogonal matrix is a square matrix such that

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Answer

Notice that this also means that the transpose of an orthogonal matrix is its inverse.

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Question

Assume M is an orthogonal matrix. Which of the following is not always true?

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Answer

Let us examine each of the options:

This is the definition of an orthogonal matrix; it is always true.

This can be directly proved from the previous statment. If you subtitute the inverse for the transpose in the definition equation, it is still true.

The determinant of any orthogonal matrix is either 1 or -1. This statment can be proved in the following way:

The incorrect statment is . Consider an example matrix:

$\textbf{M}=\begin{bmatrix} 0& -1\ 1&0 \end{bmatrix}$

which has a transpose

M and its transpose are clearly not equal. However, if we multiply them, we can see that their product is the identity matrix and they are therefore orthogonal.

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Question

The matrix M given below is orthogonal. What is x?

$\textbf{M}=\begin{bmatrix} x& -1\ 1&0 \end{bmatrix}$

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Answer

We know that for any orthogonal matrix:

So, we can set up an equation with our matrix. First, let's find the transpose of M:

$\textbf{M}=\begin{bmatrix} x& -1\ 1&0 \end{bmatrix} $\rightarrow\textbf{M}^T$ =\begin{bmatrix} x& 1\ -1& 0 \end{bmatrix}$

Now, let's set up the equation based on the definition:

Comparing the last two matricies, one can see that x=0.

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Question

The matrix A is given below. Is it orthogonal?

$\textbf{A}=\begin{bmatrix} 2& 1&0\ -1& 3&1 \ 1&0 &-1 \end{bmatrix}$

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Answer

For a matrix M to be orthogonal, it has to satisfy the following condition:

We can find the transpose of A and multiply it by A to determine whether or not it is orthogonal:

$\textbf{A}=\begin{bmatrix} 2& 1&0\ -1& 3&1 \ 1&0 &-1 \end{bmatrix} \rightarrow $\textbf{A}^T$= \begin{bmatrix} 2& -1&1\ 1& 3&0 \ 0&1&-1 \end{bmatrix}$

Therefore, A is not an orthogonal matrix.

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Question

The matrix B is given below. Is B orthogonal? (Round to three decimal places)

$\textbf{B}=\begin{bmatrix} 0.5257&-0.8507 \ 0.8507& 0.5257 \end{bmatrix}$

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Answer

For a matrix M to be orthogonal, it has to satisfy the following condition:

We can find the transpose of B and multiply it by B to determine whether or not it is orthogonal:

$\textbf{B}=\begin{bmatrix} 0.5257&-0.8507 \ 0.8507& 0.5257 \end{bmatrix}\rightarrow $\textbf{B}^T$=\begin{bmatrix} 0.5257&0.8507 \ -0.8507& 0.5257 \end{bmatrix}$

$\textbf{B} $\textbf{B}^T$=\begin{bmatrix} 0.5257&-0.8507 \ 0.8507& 0.5257 \end{bmatrix}\begin{bmatrix} 0.5257&0.8507 \ -0.8507& 0.5257 \end{bmatrix}= \begin{bmatrix} 1.000&0 \ 0& 1.000 \end{bmatrix}=\textbf{I}$

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