212 CHAPTER 11. MATRICES AND THE INNER PRODUCT

18. Find the eigenvalues and an orthonormal basis of eigenvectors for A.

A =

 11 −1 −4−1 11 −4−4 −4 14

 .

Hint: Two eigenvalues are 12 and 18.

19. Find the eigenvalues and an orthonormal basis of eigenvectors for A.

A =

 4 1 −21 4 −2−2 −2 7

 .

Hint: One eigenvalue is 3.

20. 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

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

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.

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

22. Show that if A is an Hermitian matrix and λ and µ are two different eigenvalues, thenif 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

λ (x,x) = (Ax,x) = (x,Ax) = (x,λx) = λ (x,x)

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

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

24. 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.