13.21. EXERCISES 383
−∑i, j
λ+i µ
+j (w j,vi)viw
∗j −∑
i, jλ+i µ
+j (vi,w j)w jv
∗i
]The trace satisfies trace(AB) = trace(BA) when both products make sense. Therefore,
trace(viw
∗j)= trace
(w∗jvi
)=w∗jvi ≡ (vi,w j) ,
a similar formula for w jv∗i . Therefore, this equals
= ∑i
(λ+i)2
+∑j
(µ+j
)2−2∑
i, jλ+i µ
+j
∣∣(vi,w j)∣∣2 . (13.41)
Since these are orthonormal bases,
∑i
∣∣(vi,w j)∣∣2 = 1 = ∑
j
∣∣(vi,w j)∣∣2
and so 13.41 equals
= ∑i
∑j
((λ+i)2
+(
µ+j
)2−2λ
+i µ
+j
)∣∣(vi,w j)∣∣2 .
Similarly,
|A−B|2 = ∑i
∑j
((λ i)
2 +(
µ j
)2−2λ iµ j
)∣∣(vi,w j)∣∣2 .
Now it is easy to check that (λ i)2 +(
µ j
)2−2λ iµ j ≥
(λ+i)2
+(
µ+j
)2−2λ
+i µ
+j . ■
13.21 Exercises1. Show (A∗)∗ = A and (AB)∗ = B∗A∗.
2. Prove Corollary 13.13.10.
3. Show that if A is an n×n matrix which has an inverse then A+ = A−1.
4. Using the singular value decomposition, show that for any square matrix A, it followsthat A∗A is unitarily similar to AA∗.
5. Let A,B be a m×n matrices. Define an inner product on the set of m×n matrices by
(A,B)F ≡ trace(AB∗) .
Show this is an inner product satisfying all the inner product axioms. Recall for M ann×n matrix, trace(M)≡ ∑
ni=1 Mii. The resulting norm, ||·||F is called the Frobenius
norm and it can be used to measure the distance between two matrices.
6. It was shown that a matrix A is normal if and only if it is unitarily similar to a diagonalmatrix. It was also shown that if a matrix is Hermitian, then it is unitarily similar to areal diagonal matrix. Show the converse of this last statement is also true. If a matrixis unitarily similar to a real diagonal matrix, then it is Hermitian.