Can Eigenvectors Diagonalize a Matrix with Only One Eigenvalue?

In summary, the Primary Decomposition Theorem is a fundamental result in commutative algebra that states every ideal in a Noetherian ring can be written as an intersection of primary ideals. It is important because it allows for the simplification of complex ideals into simpler components, known as primary ideals, making it easier to study and understand the structure of ideals in commutative rings. Primary decomposition refers to the process of breaking down an ideal into primary ideals, while the Primary Decomposition Theorem is the statement that guarantees this process is always possible. Primary ideals are "almost prime" ideals that capture the behavior of prime ideals in a ring. In algebraic geometry, the Primary Decomposition Theorem is used to study the structure of varieties and
  • #1
b00tofuu
11
0
Let F= R or C, and A =
[1 2 3] is considered as linear operator in F3
[0 1 2]
[0 0 1]
then the minimal polynomial of A = (x-1)^3, can we say that the primary decomposition thm doesn't give any decomposition, can we find an invertible P s.t P^-1*A*p is a block diagonal matrix?
 
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  • #2
it seems 1 is the only eigenvalue. now look for the eigenvectors with that eigenvalue. how many do you find? is that enough to diagonalize the matrix?
 

FAQ: Can Eigenvectors Diagonalize a Matrix with Only One Eigenvalue?

What is the Primary Decomposition Theorem?

The Primary Decomposition Theorem is a fundamental result in commutative algebra that states that every ideal in a Noetherian ring can be written as an intersection of primary ideals. It is a powerful tool in studying the structure of ideals in commutative rings.

Why is the Primary Decomposition Theorem important?

The Primary Decomposition Theorem is important because it allows us to break down complex ideals into simpler components, known as primary ideals. This makes it easier to study and understand the structure of ideals in commutative rings, leading to a deeper understanding of the ring itself.

What is the difference between primary decomposition and primary decomposition theorem?

Primary decomposition refers to the process of breaking down an ideal into primary ideals, while the Primary Decomposition Theorem is the statement that guarantees that this process is always possible for ideals in a Noetherian ring. In other words, primary decomposition is a method, while the Primary Decomposition Theorem is a result.

What are primary ideals?

Primary ideals are ideals in a commutative ring that have a special property - if the product of two elements in the ring is in the ideal, then at least one of the elements must be in the ideal. In other words, primary ideals are "almost prime" ideals that capture the behavior of prime ideals in a ring.

How is the Primary Decomposition Theorem used in algebraic geometry?

In algebraic geometry, the Primary Decomposition Theorem is used to study the structure of varieties and their ideals. By decomposing an ideal into primary ideals, we can understand the irreducible components of a variety and their intersections. This allows us to analyze and classify algebraic varieties, leading to important results in the field.

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