326 CHAPTER 13. MATRICES AND THE INNER PRODUCT

15. Show that if A is a real symmetric matrix and λ and µ are two different eigenvalues,then if x is an eigenvector for λ and y is an eigenvector for µ, then x ·y = 0. Also alleigenvalues are real. Supply reasons for each step in the following argument. First

λxT x = (Ax)T x = xT Ax = xT Ax = xTλx = λxT x

and so λ = λ . This shows that all eigenvalues are real. It follows all the eigenvectorsare real. Why? Now let x,y,µ and λ be given as above.

λ (x ·y) = λx ·y = Ax ·y = x ·Ay = x·µy = µ (x ·y) = µ (x ·y)

and so(λ −µ)x ·y = 0.

Since λ ̸= µ, it follows x ·y = 0.

16. Suppose U is an orthogonal n×n matrix. Explain why rank(U) = n.

17. Show that if A is an Hermitian matrix and λ and µ are two different eigenvalues,then if x is an eigenvector for λ and y is an eigenvector for µ, then x ·y = 0. Also alleigenvalues are real. Supply reasons for each step in the following argument. First

λx ·x = Ax ·x = x·Ax = x·λx = λx ·x

and so λ = λ . This shows that all eigenvalues are real. Now let x,y,µ and λ be givenas above.

λ (x ·y) = λx ·y = Ax ·y = x ·Ay = x·µy = µ (x ·y) = µ (x ·y)

and so (λ −µ)x ·y = 0. Since λ ̸= µ, it follows x ·y = 0.

18. Show that the eigenvalues and eigenvectors of a real matrix occur in conjugate pairs.

19. If a real matrix A has all real eigenvalues, does it follow that A must be symmetric.If so, explain why and if not, give an example to the contrary.

20. Suppose A is a 3×3 symmetric matrix and you have found two eigenvectors whichform an orthonormal set. Explain why their cross product is also an eigenvector.

21. Study the definition of an orthonormal set of vectors. Write it from memory.

22. Determine which of the following sets of vectors are orthonormal sets. Justify youranswer.

(a) {(1,1) ,(1,−1)}

(b){(

1√2, −1√

2

),(1,0)

}(c)

{( 13 ,

23 ,

23

),(−2

3 , −13 , 2

3

),( 2

3 ,−23 , 1

3

)}23. Show that if {u1, · · · ,un} is an orthonormal set of vectors in Fn, then it is a basis.

Hint: It was shown earlier that this is a linearly independent set. If you wish, replaceFn with Rn. Do this version if you do not know the dot product for vectors in Cn.