Q 5.5-5E Question: Is 鈱〢,B鈱�=trace(A,B... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q 5.5-5E (page 260)

Question: Is $鈱� 鈱�A, B鈱� 鈱� = t r a c e (A, B^{T})$ an inner product in R^苍脳尘? (The notation $鉄� 鈱� A, B 鈱� 鉄�$ is chosen to distinguish this product from the one considered in Example 3 and Exercise 4.)

Short Answer

Expert verified

$鈱�鈱�,鈱�鈱�$ is an inner product.

Step by step solution

Inner product of Rn×m

Observe the following conditions.

${(A A^{T})}_{i i} = {鈭�}_{k = 1}^{m} A_{i k} {(A^{T})}_{i k} = {鈭�}_{k = 1}^{m} a_{i k} a_{i k} = {鈭�}_{k = 1}^{m} {a^{2}}_{i k} = \begin{array}{l} 鈱�鈱�A, A鈱�鈱� = t r a c e (A A^{T}) \\ = {鈭�}_{i = 1}^{n} {鈭�}_{k = 1}^{m} {a^{2}}_{i k} \end{array}$

Thus, $鈱�鈱�A, A鈱�鈱� > 0$ if and only if $A 鈮� 0$ and is equals to zero if $A = 0$

$\begin{array}{l} 鈱�鈱�A, B鈱�鈱� = t r a c e (A B^{T}) \\ = t r a c e ({(A B^{T})}^{T}) \\ = t r a c e (B A^{T}) \\ = 鈱�鈱�B, A鈱�鈱� \end{array}$

$\begin{array}{l} 鈱�鈱�伪 A, B鈱�鈱� = t r a c e (伪 A B^{T}) \\ = 伪 t r a c e (A B^{T}) \\ = 伪鈱�鈱�B, A鈱�鈱� \end{array}$

$\begin{array}{l} 鈱�鈱�A + B, C鈱�鈱� = t r a c e ((A + B) C^{T}) \\ = t r a c e (A C^{T} + B C^{T}) \\ = t r a c e (A C^{T}) + t r a c e (B C^{T}) \\ = 鈱�鈱�A, B鈱�鈱� + 鈱�鈱�B, C鈱�鈱� \end{array}$

Hence, $鈱�鈱�,鈱�鈱�$ is an inner product on R^苍脳尘.

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91影视

Question: Is $鈱� 鈱�A, B鈱� 鈱� = t r a c e (A, B^{T})$ an inner product in R^苍脳尘? (The notation $鉄� 鈱� A, B 鈱� 鉄�$ is chosen to distinguish this product from the one considered in Example 3 and Exercise 4.)

Short Answer

Step by step solution

Inner product of Rn×m

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91影视

Question: Is 鈱�A,B鈱�=trace(A,BT) an inner product in R苍脳尘? (The notation鉄�鈱�A,B鈱�鉄� is chosen to distinguish this product from the one considered in Example 3 and Exercise 4.)

Short Answer

Step by step solution

Inner product of Rn×m

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Mechanics Maths

Pure Maths

Statistics

Calculus

Geometry

Study anywhere. Anytime. Across all devices.

Question: Is $鈱� 鈱�A, B鈱� 鈱� = t r a c e (A, B^{T})$ an inner product in R^苍脳尘? (The notation $鉄� 鈱� A, B 鈱� 鉄�$ is chosen to distinguish this product from the one considered in Example 3 and Exercise 4.)