3. Let T: R³ R³ be the operator given by 00 8 1 0 T(v) = -12 6 υ. Determine whether T is decomposable or indecomposable.
3. Let T: R³ R³ be the operator given by 00 8 1 0 T(v) = -12 6 υ. Determine whether T is decomposable or indecomposable.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![### Matrix Operator Application and Properties
**3. Given Operator:**
Let \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) be the operator defined by the matrix:
\[ T(v) = \begin{pmatrix}
0 & 0 & 8 \\
1 & 0 & -12 \\
0 & 1 & 6
\end{pmatrix} v \]
**Task:**
Determine whether \( T \) is decomposable or indecomposable.
---
**4. Cyclic Operator Assumption:**
Assume \( S \) is a cyclic operator on the finite-dimensional vector space \( U \).
---
### Detailed Explanation:
The provided text outlines a mathematical problem related to matrix operators in the field of linear algebra. Specifically, the task is to determine the structural property of the operator \( T \) given by a \( 3 \times 3 \) matrix.
#### Matrix Representation:
The operator \( T \) is represented by the following matrix:
\[ \begin{pmatrix}
0 & 0 & 8 \\
1 & 0 & -12 \\
0 & 1 & 6
\end{pmatrix} \]
This matrix transforms a vector \( v \in \mathbb{R}^3 \) by matrix multiplication.
#### Concept of Decomposability:
In linear algebra, an operator (or matrix) is **decomposable** if it can be written as a direct sum of two non-trivial subspaces. Otherwise, it is **indecomposable**.
---
Next:
- Analyze the matrix for properties such as eigenvalues, eigenvectors, and invariant subspaces.
- Use these properties to determine if the operator \( T \) can be decomposed into simpler parts.
For further understanding, consider concepts like Jordan canonical forms, cyclic subspaces, and the theory of linear transformations.
---
Visit our [Linear Algebra Section](#) for more detailed explanations and examples on matrix operators, their properties, and applications.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fae19e8bd-5915-473c-8a03-9ac053dffb7e%2F84ae3e58-e2f9-4076-aaf3-e6588520c2d3%2Ff6rr7b7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Matrix Operator Application and Properties
**3. Given Operator:**
Let \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) be the operator defined by the matrix:
\[ T(v) = \begin{pmatrix}
0 & 0 & 8 \\
1 & 0 & -12 \\
0 & 1 & 6
\end{pmatrix} v \]
**Task:**
Determine whether \( T \) is decomposable or indecomposable.
---
**4. Cyclic Operator Assumption:**
Assume \( S \) is a cyclic operator on the finite-dimensional vector space \( U \).
---
### Detailed Explanation:
The provided text outlines a mathematical problem related to matrix operators in the field of linear algebra. Specifically, the task is to determine the structural property of the operator \( T \) given by a \( 3 \times 3 \) matrix.
#### Matrix Representation:
The operator \( T \) is represented by the following matrix:
\[ \begin{pmatrix}
0 & 0 & 8 \\
1 & 0 & -12 \\
0 & 1 & 6
\end{pmatrix} \]
This matrix transforms a vector \( v \in \mathbb{R}^3 \) by matrix multiplication.
#### Concept of Decomposability:
In linear algebra, an operator (or matrix) is **decomposable** if it can be written as a direct sum of two non-trivial subspaces. Otherwise, it is **indecomposable**.
---
Next:
- Analyze the matrix for properties such as eigenvalues, eigenvectors, and invariant subspaces.
- Use these properties to determine if the operator \( T \) can be decomposed into simpler parts.
For further understanding, consider concepts like Jordan canonical forms, cyclic subspaces, and the theory of linear transformations.
---
Visit our [Linear Algebra Section](#) for more detailed explanations and examples on matrix operators, their properties, and applications.
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