13.5. EXERCISES 341

15. Give some disks in the complex plane whose union contains all the eigenvalues ofthe matrix  1+2i 4 2

0 i 35 6 7

16. Show a square matrix is invertible if and only if it has no zero eigenvalues.

17. Using Schur’s theorem, show the trace of an n× n matrix equals the sum of theeigenvalues and the determinant of an n×n matrix is the product of the eigenvalues.

18. Using Schur’s theorem, show that if A is any complex n× n matrix having eigen-values {λ i} listed according to multiplicity, then ∑i, j

∣∣Ai j∣∣2 ≥ ∑

ni=1 |λ i|2. Show that

equality holds if and only if A is normal. This inequality is called Schur’s inequality.[33]

19. Here is a matrix. 1234 6 5 3

0 −654 9 12398 123 10,000 1156 78 98 400

I know this matrix has an inverse before doing any computations. How do I know?

20. Show the critical points of the following function are

(0,−3,0) ,(2,−3,0) ,and(

1,−3,−13

)and classify them as local minima, local maxima or saddle points.

f (x,y,z) =− 32 x4 +6x3−6x2 + zx2−2zx−2y2−12y−18− 3

2 z2.

21. Here is a function of three variables.

f (x,y,z) = 13x2 +2xy+8xz+13y2 +8yz+10z2

change the variables so that in the new variables there are no mixed terms, termsinvolving xy,yz etc. Two eigenvalues are 12 and 18.

22. Here is a function of three variables.

f (x,y,z) = 2x2−4x+2+9yx−9y−3zx+3z+5y2−9zy−7z2

change the variables so that in the new variables there are no mixed terms, termsinvolving xy,yz etc. The eigenvalues of the matrix which you will work with are− 17

2 , 192 ,−1.

23. Here is a function of three variables.

f (x,y,z) =−x2 +2xy+2xz− y2 +2yz− z2 + x

change the variables so that in the new variables there are no mixed terms, termsinvolving xy,yz etc.