The following is one way to define the quaternions, discovered in 1843 by the Irish mathematician Sir W. R. Hamilton. Consider the set H of all 4 x 4 matrices M of the form M = P 9 T S -q -T -S P S -T -8 P 9 T -9 P where p, q, r, s are arbitrary real numbers. We can write M more succinctly in partitioned form as M = A-B BAT where A and B are rotation-scaling matrices. (a) Show that H is closed under addition: If M and N are in H, then so is M + N. (b) Show that H is closed under scalar multiplication: If M is in H and k is an arbitrary scalar, then kM is in H. (c) The above show that H is a subspace of the linear space R4×4. Find a basis of H, and thus determine the dimension of H.
The following is one way to define the quaternions, discovered in 1843 by the Irish mathematician Sir W. R. Hamilton. Consider the set H of all 4 x 4 matrices M of the form M = P 9 T S -q -T -S P S -T -8 P 9 T -9 P where p, q, r, s are arbitrary real numbers. We can write M more succinctly in partitioned form as M = A-B BAT where A and B are rotation-scaling matrices. (a) Show that H is closed under addition: If M and N are in H, then so is M + N. (b) Show that H is closed under scalar multiplication: If M is in H and k is an arbitrary scalar, then kM is in H. (c) The above show that H is a subspace of the linear space R4×4. Find a basis of H, and thus determine the dimension of H.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![The following is one way to define the quaternions, discovered in 1843 by the Irish mathematician Sir W. R. Hamilton. Consider the set \( H \) of all \( 4 \times 4 \) matrices \( M \) of the form
\[
M = \begin{bmatrix}
p & -q & -r & -s \\
q & p & s & -r \\
r & -s & p & q \\
s & r & -q & p
\end{bmatrix}
\]
where \( p, q, r, s \) are arbitrary real numbers. We can write \( M \) more succinctly in partitioned form as
\[
M = \begin{bmatrix}
A & -B^T \\
B & A^T
\end{bmatrix}
\]
where \( A \) and \( B \) are rotation–scaling matrices.
(a) Show that \( H \) is closed under addition: If \( M \) and \( N \) are in \( H \), then so is \( M + N \).
(b) Show that \( H \) is closed under scalar multiplication: If \( M \) is in \( H \) and \( k \) is an arbitrary scalar, then \( kM \) is in \( H \).
(c) The above show that \( H \) is a subspace of the linear space \( \mathbb{R}^{4 \times 4} \). Find a basis of \( H \), and thus determine the dimension of \( H \).
(d) Show that \( H \) is closed under multiplication: If \( M \) and \( N \) are in \( H \), then so is \( MN \).
(e) Show that if \( M \) is in \( H \), then so is \( M^T \).
(f) For a matrix \( M \) in \( H \), compute \( M^T M \).
(g) Which matrices \( M \) in \( H \) are invertible? If a matrix \( M \) in \( H \) is invertible, is \( M^{-1} \) necessarily in \( H \) as well?
(h) If \( M \) and \( N \) are in \( H \), does the equation \( MN = NM \) always hold?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5fafa0a8-e885-4f45-9f8c-34c617df8f38%2F5817503b-b9f0-41d0-a82e-e2495e2e2157%2Fcwvwvb_processed.png&w=3840&q=75)
Transcribed Image Text:The following is one way to define the quaternions, discovered in 1843 by the Irish mathematician Sir W. R. Hamilton. Consider the set \( H \) of all \( 4 \times 4 \) matrices \( M \) of the form
\[
M = \begin{bmatrix}
p & -q & -r & -s \\
q & p & s & -r \\
r & -s & p & q \\
s & r & -q & p
\end{bmatrix}
\]
where \( p, q, r, s \) are arbitrary real numbers. We can write \( M \) more succinctly in partitioned form as
\[
M = \begin{bmatrix}
A & -B^T \\
B & A^T
\end{bmatrix}
\]
where \( A \) and \( B \) are rotation–scaling matrices.
(a) Show that \( H \) is closed under addition: If \( M \) and \( N \) are in \( H \), then so is \( M + N \).
(b) Show that \( H \) is closed under scalar multiplication: If \( M \) is in \( H \) and \( k \) is an arbitrary scalar, then \( kM \) is in \( H \).
(c) The above show that \( H \) is a subspace of the linear space \( \mathbb{R}^{4 \times 4} \). Find a basis of \( H \), and thus determine the dimension of \( H \).
(d) Show that \( H \) is closed under multiplication: If \( M \) and \( N \) are in \( H \), then so is \( MN \).
(e) Show that if \( M \) is in \( H \), then so is \( M^T \).
(f) For a matrix \( M \) in \( H \), compute \( M^T M \).
(g) Which matrices \( M \) in \( H \) are invertible? If a matrix \( M \) in \( H \) is invertible, is \( M^{-1} \) necessarily in \( H \) as well?
(h) If \( M \) and \( N \) are in \( H \), does the equation \( MN = NM \) always hold?
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