Orthogonal Matrices - Linear Algebra

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Question

Determine if the following matrix is orthogonal or not.

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

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.

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

$A.A^T=\begin{bmatrix} (-1)(-1)&(0)(0) \ (0)(0)& (1)(1) \end{bmatrix}=\begin{bmatrix} 1&0\ 0& 1 \end{bmatrix}$

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?

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

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?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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?

Answer

Let us examine each of the options:

$\textbf{M}^T\textbf{M}=\textbf{I}$ This is the definition of an orthogonal matrix; it is always true.

$\textbf{M}^T=\textbf{M}^{-1}$ 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.

$(\textup{det}(\textbf{M}))^2=1$ The determinant of any orthogonal matrix is either 1 or -1. This statment can be proved in the following way:

$1=\textup{det}(\textbf{I})=\textup{det}(\textbf{M}^T \textbf{M}) =\textup{det}(\textbf{M}^T)\textup{det}(\textbf{M})=(\textup{det}(\textbf{M}))^2$

The incorrect statment is $\textbf{M}=\textbf{M}^T$ . Consider an example matrix:

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

which has a transpose

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

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.

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

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Question

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

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

Answer

We know that for any orthogonal matrix:

$\textbf{M}\textbf{M}^T=\textbf{I}$

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:

$\textbf{MM}^T=\begin{bmatrix} x& -1\ 1&0 \end{bmatrix} \begin{bmatrix} x& 1\ -1& 0 \end{bmatrix} =\begin{bmatrix} x^2+1& x\ x&1 \end{bmatrix}=\begin{bmatrix} 1&0 \ 0& 1 \end{bmatrix}=\textbf{I}$

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?

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

Answer

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

$\textbf{MM}^T=\textbf{I}$

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

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

$\textbf{AA}^T= \begin{bmatrix} 2& 1&0\ -1& 3&1 \ 1&0 &-1 \end{bmatrix} \begin{bmatrix} 2& -1&1\ 1& 3&0 \ 0&1&-1 \end{bmatrix}= \begin{bmatrix} 5& 5&-2 \ 5&11 &-2 \ -2& -2& 2 \end{bmatrix} eq \begin{bmatrix} 1& 0& 0\ 0& 1& 0\ 0& 0&1 \end{bmatrix}$

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)

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

Answer

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

$\textbf{MM}^T=\textbf{I}$

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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