Using paper and pencil, find the QR factorization of the matrices in Exercises 15 through 28. Compare with Exercises 1 through 14.
The QR factorization of the matrix is .
Consider the matrix where and
By the theorem of QR method, the value of and is defined as follows:
Simplify the equation as follows:
Substitute the values 7 for and for in the equation as follows:
Therefore, the values localid="1659437134701" .
As , substitute the values for and for in the equation
localid="1659437594586" as follows:
Substitute the values for ,3 for and for in the equation as follows:
The value of and is defined as follows.
Simplify the equation localid="1659438712674" as follows.
Substitute the values for and for in the equation as follows.
The values localid="1659439047262" and
Therefore, the matrices and .
Hence, the QR factorization of the matrix is localid="1659439350112" .
This exercise shows one way to define the quaternions,discovered in 1843 by the Irish mathematician Sir W.R. Hamilton (1805-1865).Consider the set H of all matrices M of the form
where p,q,r,s are arbitrary real numbers.We can write M more sufficiently in partitioned form as
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
c.Parts (a) and (b) Show that H is a subspace of the linear space .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 .
f.For a matrix M in H compute .
g.Which matrices M in H are invertible.If a matrix M in H is invertible is necessarily in H as well?
h. If M and N are in H,does the equation always hold?
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