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Unformatted text preview: errors are about the same. In general, the two errors are related as
follows. Let x = 0, x = 0, and
˜
ǫ= x −x
˜
,
x ǫ=
˜ x −x
˜
.
x
˜ If ǫ < 1, then ǫ
ǫ
≤ǫ≤
˜
.
1+ǫ
1−ǫ
This follows from ǫ = ǫ x / x and 1 − ǫ ≤ x / x ≤ 1 + ǫ .
˜
˜
˜
˜
˜ Exercises
√
(i) Let x ∈ Cn . Prove: x 2 ≤
x 1 x ∞.
(ii) For each equality below, determine a class of vectors that satisfy the equality:
√
x 1 = x ∞,
x 1 = n x ∞,
x 2 = x ∞,
x 2 = n x ∞.
(iii) Give examples of vectors x , y ∈ Cn with x ∗ y = 0 for which
x ∗ y  = x 1 y ∞ . Also ﬁnd examples for x ∗ y  = x 2 y 2 .
(iv) The pnorm of a vector does not change when the vector is permuted.
Prove: If P is a permutation matrix, then P x p = x p .
(v) The two norm of a vector does not change when the vector is multiplied by
a unitary matrix.
Prove: If the matrix V ∈ Cn×n is unitary, then V x 2 = x 2 for any vector
x ∈ Cn .
(vi) Prove: If Q ∈ Cn×n is unitary and x ∈ Cn is a nonzero vector with Qx = λx ,
where λ is a scalar, then λ = 1.
1. Verify that the vector pnorms do indeed satisfy the three properties of a
vector norm in Deﬁnition 2.6. Downloaded 01/17/14 to 143.215.200.123. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 2.6. Matrix Norms 33 2. Reverse Triangle Inequality.
Let x , y ∈ Cn and let · be a vector norm. Prove: x − y ≤ x − y .
3. Theorem of Pythagoras.
Prove: If x , y ∈ Cn and x ∗ y = 0, then x ± y 2 = x 2 + y 2 .
2
2
2
4. Parallelogram Equality.
Let x , y ∈ Cn . Prove: x + y 2 + x − y 2 = 2( x 2 + y 2 ).
2
2
2
2
5. Polarization Identity.
Let x , y ∈ Cn . Prove: ℜ(x ∗ y) = 1 ( x + y 2 − x − y 2 ), where ℜ(α) is the
2
2
4
real part of a complex number α .
6. Let x ∈ Cn . Prove:
√
x 2 ≤ x 1 ≤ n x 2,
√
x ∞ ≤ x 2 ≤ n x ∞,
x ∞ ≤ x 1 ≤ n x ∞.
7. Let A ∈ Cn×n be nonsingular. Show that x A = Ax p is a vector norm. 2.6 Matrix Norms
We need to separate matrices from vectors inside the norms. To see this, let...
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This note was uploaded on 01/30/2014 for the course MATH 1502 taught by Professor Mcclain during the Spring '07 term at Georgia Institute of Technology.
 Spring '07
 McClain
 Multiplication, Scalar

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