Iii give examples of vectors x y cn with x y 0 for

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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 p-norm 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 p-norms 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.

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