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Chapter 5: Q3E (page 263)

TRUE OR FALSE? If matrix is A orthogonal, then the matrixA2 must be orthogonal as well.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Recollect the properties of the orthogonal a matrix.

According to the properties of a orthogonal matrix, a n x nmatrix Ais an orthogonal matrix, if ||Ax鈫�||=||x鈫�||for allx鈫�鈭�Rn.

02

Check whether the given statement is a true or false.

The given statement is if the matrix A is orthogonal, then the matrixA2 must be orthogonal as well.

Here, A is orthogonal. This means,Ax鈫�=x鈫� .

Then,

role="math" localid="1661256896384" A2x鈫�=AAx鈫�=Ax鈫�=x鈫�

It is clear that, the given statement is true, by the properties of orthogonal matrices.

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Most popular questions from this chapter

Find the angle 胃between each of the pairs of vectors u鈬赌and v鈬赌in exercises 4 through 6.

4. u鈬赌=[11],v鈬赌=[711].

Among all the vectors in Rnwhose components add up to 1, find the vector of minimal length. In the casen=2 , explain your solution geometrically.

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 4脳4matrices M of the form

M=[p鈭�q鈭�r鈭�sqps鈭�rr鈭�spqsr鈭�qp]

where p,q,r,s are arbitrary real numbers.We can write M more sufficiently in partitioned form as

M=(A鈭�BTBAT)

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 isM+N

M+Nb.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.Parts (a) and (b) Show that H is a subspace of the linear space R4脳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 MT.

f.For a matrix M in H compute MTM.

g.Which matrices M in H are invertible.If a matrix M in H is invertible isM鈭�1 necessarily in H as well?

h. If M and N are in H,does the equationMN=NMalways hold?

If then脳nmatrices Aand Bare orthogonal, which of the matrices in Exercise 5 through 11 must be orthogonal as well?3A.

If then脳n matrices Aand Bare orthogonal, which of the matrices in Exercise 5 through 11 must be orthogonal as well?B-1.
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