Group theory? This solution doesn't make sense....

In summary, the conversation discusses the relationship between matrices A and B, where A is proportional to the identity matrix. It is mentioned that there is a transform that can make both A and B diagonal, but it does not necessarily mean that B is diagonalizable. The key point is that A and B must commute for this transform to work. Additionally, in the case where A is proportional to the identity matrix, it will always be diagonal.
  • #1
applestrudle
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Case 2:
I get that D = c I means A must also be proportional to I but how does that mean B must be diagonal?

Question:
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Answers:
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  • #2
It does not show that B is diagonalizable. B is presumed to be such. The point is that there is a transform that makes B diagonal and also makes A diagonal IFF A and B commute. The particular case of A proportional to the identity matrix means that A is always diagonal. Because every similarity transform on the identity matrix simply gives back the identity matrix. So, in this case, any similarity transform that makes B diagonal will leave A unchanged and still diagonal.
 
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  • #3
DEvens said:
It does not show that B is diagonalizable. B is presumed to be such. The point is that there is a transform that makes B diagonal and also makes A diagonal IFF A and B commute. The particular case of A proportional to the identity matrix means that A is always diagonal. Because every similarity transform on the identity matrix simply gives back the identity matrix. So, in this case, any similarity transform that makes B diagonal will leave A unchanged and still diagonal.
THANKS! :D
 

1. What is group theory?

Group theory is a branch of mathematics that studies the structure and properties of groups, which are mathematical structures used to describe and analyze symmetry, patterns, and transformations.

2. What are the main applications of group theory?

Group theory has a wide range of applications in mathematics, physics, chemistry, and computer science. It is used to solve problems in abstract algebra, cryptography, particle physics, crystallography, and many other fields.

3. How is group theory related to symmetry?

Group theory is intimately connected to symmetry, as groups are used to describe and classify symmetries in various objects and systems. The concepts and techniques of group theory are essential in understanding and analyzing the symmetries of geometric shapes, molecules, crystals, and physical laws.

4. What are the basic elements of a group?

A group is made up of a set of elements and a binary operation that combines any two elements to produce a third element in the group. The elements must satisfy four axioms: closure, associativity, identity, and invertibility. These axioms ensure that the group is a well-behaved mathematical structure.

5. How is group theory used in chemistry?

In chemistry, group theory is used to classify molecules based on their symmetries and to predict their physical and chemical properties. It provides a powerful tool for understanding the behavior of molecules and their interactions, and it is particularly useful in the study of molecular orbitals and spectroscopy.

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