Additional Exercises for Chapter 9: Orthogonality

Find an orthonormal basis for the span of each of the following sets of vectors.

(a)

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(b)

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(c)

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Using the Gram Schmidt process find an orthonormal basis for the following span:

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A solution is

Using the Gram Schmidt process find an orthonormal basis for the following span:

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A solution is

The set is a subspace of Find an orthonormal basis for this subspace.
The subspace is of the form

So a basis is . Therefore, an orthonormal basis is

In each case, write as the sum of a vector in and a vector in .
,

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Normalize the rows to make each of the following matrices orthogonal.

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, not both zero

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Find the eigenvalues and an orthonormal basis of eigenvectors for
Two eigenvalues are 12 and 18.

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The eigenvectors and eigenvalues are:

Find the eigenvalues and an orthonormal basis of eigenvectors for
One eigenvalue is 3.

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The eigenvectors and eigenvalues are:

Find the eigenvalues and an orthonormal basis of eigenvectors for Diagonalize by finding an orthogonal matrix and a diagonal matrix such that .
One eigenvalue is .

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The eigenvectors and eigenvalues are:

Find the eigenvalues and an orthonormal basis of eigenvectors for Diagonalize by finding an orthogonal matrix and a diagonal matrix such that .
Two eigenvalues are and .

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The eigenvectors and eigenvalues are: The matrix has these as its columns.

Find the eigenvalues and an orthonormal basis of eigenvectors for Diagonalize by finding an orthogonal matrix and a diagonal matrix such that .
Two eigenvalues are and 18.

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The eigenvectors and eigenvalues are: The matrix has these as its columns.

Find the eigenvalues and an orthonormal basis of eigenvectors for Diagonalize by finding an orthogonal matrix and a diagonal matrix such that .
The eigenvalues are

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eigenvectors: These vectors are the columns of .

Find the eigenvalues and an orthonormal basis of eigenvectors for Diagonalize by finding an orthogonal matrix and a diagonal matrix such that . 

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The eigenvectors and eigenvalues are: These vectors are the columns of the matrix .

Find the eigenvalues and an orthonormal basis of eigenvectors for Diagonalize by finding an orthogonal matrix and a diagonal matrix such that .

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The eigenvectors and eigenvalues are: These vectors are the columns of .

Find the eigenvalues and an orthonormal basis of eigenvectors for Diagonalize by finding an orthogonal matrix and a diagonal matrix such that .
The eigenvalues are where is listed twice because it is a root of multiplicity 2.

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The eigenvectors and eigenvalues are: The columns are these vectors.

Find the eigenvalues and an orthonormal basis of eigenvectors for Diagonalize by finding an orthogonal matrix and a diagonal matrix such that .
The eigenvalues are

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The eigenvectors and eigenvalues are: The columns are these vectors.

Find the eigenvalues and an orthonormal basis of eigenvectors for the matrix
The eigenvalues are

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The eigenvectors: Then the columns of are these vectors

Find the eigenvalues and an orthonormal basis of eigenvectors for the matrix
The eigenvalues are where is listed twice because it has multiplicity 2 as a zero of the characteristic equation.

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The eigenvectors and eigenvalues are: The columns of are these vectors.

Explain why a matrix is symmetric if and only if there exists an orthogonal matrix such that for a diagonal matrix.
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If is given by the formula, then Next suppose Then by the theorems on symmetric matrices, there exists an orthogonal matrix such that for diagonal. Hence

Show that if is a real symmetric matrix and and are two different eigenvalues, then if is an eigenvector for and is an eigenvector for then Also all eigenvalues are real. Supply reasons for each step in the following argument. First and so This shows that all eigenvalues are real. It follows all the eigenvectors are real. Why? Now let and be given as above.   and so Why does it follow that

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Since it follows

Challenge Exercises

If is a subspace of , show that .
Show that , then use Theorem th:023783c twice.
If is a subspace of , show how to find an matrix such that .
Practice Problem prob:8_1_13.
Let be a basis of , and let be the matrix with rows . Then if and only if for each ; if and only if is in .
If and are subspaces, define to be the set of all possible sums of elements of with elements of . Is a subspace? Show that .
Think of as consisting of row vectors.
Let be an matrix, and let . Show that the following are equivalent.
(a)
( is a projection matrix).
(b)
for all and in .
(c)
for all in .
For (ii) implies (iii): Write and use the uniqueness argument preceding the definition of . For (iii) implies (ii): is in for all in .
If is a projection matrix, show that is also a projection matrix.
If and and are projection matrices, show that is also a projection matrix.
If is and is invertible, show that is a projection matrix.
Consider where one of . Show that the characteristic polynomial (see Definition def:char_poly_complex) is given by , where , and find an orthogonal matrix such that is diagonal.

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Consider . Show that the characteristic polynomial (see Definition def:char_poly_complex) is given by , and find an orthogonal matrix such that is diagonal.
Given , show that the characteristic polynomial (see Definition def:char_poly_complex) is given by , and find an orthogonal matrix such that is diagonal.
Consider . Show that the characteristic polynomial (see Definition def:char_poly_complex) is given by , and find an orthogonal matrix such that is diagonal.

Octave Exercises

Use Octave to check your work on Problems prb:9.1 to prb:9.12. The first steps of prb:9.2 are in the code below. See if you can finish the rest of the problem.

To use Octave, go to the Sage Math Cell Webpage, copy the code below into the cell, select OCTAVE as the language, and press EVALUATE. 

A=[4 1 -2; 1 4 -2; -2 -2 7];
 

 
[Q,D]=eig(A)
 

 
% The eigenvalue 3 has algebraic multiplicity 2 and geometric multiplicity 2
 
% One way to see this is to compute:
 
rref(A-3*eye(size(A)))
 

                                                                  

                                                                  
 
% By hand, we will not get the same eigenvectors for 3 that Octave did, and we will need to
 
% orthogonalize them using Gram-Schmidt.  Octave does so automatically.  Observe:
 
transpose(Q)*Q
 

 
% And so we can check that the following is (approximately) the same as D:
 
transpose(Q)*A*Q

Bibliography

Some of these problems come from Section 7.4 of Ken Kuttler’s A First Course in Linear Algebra. (CC-BY)

Ken Kuttler, A First Course in Linear Algebra, Lyryx 2017, Open Edition, pp. 433–438.

Other problems come from the second part of Section 8.1 of Keith Nicholson’s Linear Algebra with Applications. (CC-BY-NC-SA)

W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2018, Open Edition, p. 422–423