Q38E If B is a diagonal 4脳4聽matrix,... [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 4: Q38E (page 177)

If B is a diagonal $4 脳 4$ matrix, what are the possible dimensions of the space V of all data-custom-editor="chemistry" $4 脳 4$ matrices A that commute with B?

Short Answer

Expert verified

The possible dimensions of the space V of all $4 脳 4$ matrices A that commute with B aredata-custom-editor="chemistry" $位 = 3, 5, 6 or 9$ .

Step by step solution

Explanation for the dimension of the space

If a linear space has a basis with n elements then all other bases of V, consists of n elements as well.

Also we say that n is the dimension of V

$\dim (V) = n$

Solution for the dimension of the basis

Consider, B is a diagonal with $4 脳 4$ matrices.

Assume, $B = [\begin{matrix} 3 & 0 & 0 & 0 \\ 0 & 5 & 0 & 0 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 9 \end{matrix}]$ is the $4 脳 4$ diagonal matrix of B

$|位滨 - B| = 位 [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] - [\begin{matrix} 3 & 0 & 0 & 0 \\ 0 & 5 & 0 & 0 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 9 \end{matrix}] |位滨 - B| = 0 [\begin{matrix} 位 & 0 & 0 & 0 \\ 0 & 位 & 0 & 0 \\ 0 & 0 & 位 & 0 \\ 0 & 0 & 0 & 位 \end{matrix}] - [\begin{matrix} 3 & 0 & 0 & 0 \\ 0 & 5 & 0 & 0 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 9 \end{matrix}] = 0 [\begin{matrix} 位 - 3 & 0 & 0 & 0 \\ 0 & 位 - 5 & 0 & 0 \\ 0 & 0 & 位 - 6 & 0 \\ 0 & 0 & 0 & 位 - 9 \end{matrix}] = 0$

Further simplify as follows

data-custom-editor="chemistry" $(位 - 3) (位 - 5) (位 - 6) (位 - 9) = 0 位 = 3, 5, 6, or 9$

Thus the possible dimensions of the space V of all $4 脳 4$ matrices A that commute with B are data-custom-editor="chemistry" $位 = 3, 5, 6 or 9$ .

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91影视

If B is a diagonal $4 脳 4$ matrix, what are the possible dimensions of the space V of all data-custom-editor="chemistry" $4 脳 4$ matrices A that commute with B?

Short Answer

Step by step solution

Explanation for the dimension of the space

Solution for the dimension of the basis

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91影视

If B is a diagonal 4脳4matrix, what are the possible dimensions of the space V of all data-custom-editor="chemistry" 4脳4matrices A that commute with B?

Short Answer

Step by step solution

Explanation for the dimension of the space

Solution for the dimension of the basis

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Discrete Mathematics

Decision Maths

Geometry

Calculus

Statistics

Study anywhere. Anytime. Across all devices.

If B is a diagonal $4 脳 4$ matrix, what are the possible dimensions of the space V of all data-custom-editor="chemistry" $4 脳 4$ matrices A that commute with B?