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Magazine Chapter 4: Problem 43
Prove that if \(A\) is a diagonalizable matrix with only one eigenvalue \(\lambda\), then \(A\) is of the form \(A=\lambda L\). (Such a matrix is called a scalar matrix.
Short Answer
Expert verified
If a diagonalizable matrix has only one eigenvalue, it is a scalar matrix.
Step by step solution
01
Understanding Diagonalizable Matrices
A matrix \( A \) is diagonalizable if there exists an invertible matrix \( P \) and a diagonal matrix \( D \) such that \( A = PDP^{-1} \). The matrix \( D \) contains the eigenvalues of \( A \) on its diagonal, and the columns of \( P \) are the eigenvectors of \( A \). In this scenario, we know \( A \) is diagonalizable and it has only one eigenvalue, \( \lambda \).
02
Formulating the Diagonal Matrix
Since \( A \) has only one eigenvalue \( \lambda \), the diagonal matrix \( D \) must have \( \lambda \) on its main diagonal for all its diagonal entries. Thus, \( D = \lambda I \), where \( I \) is the identity matrix. This means all diagonal entries of \( D \) are equal to \( \lambda \).
03
Matrix Representation of Identity Matrix
The identity matrix \( I \) is a special matrix with 1s on the diagonal and 0s elsewhere. Since \( D = \lambda I \), it means \( D \) scales each 1 on the identity matrix by \( \lambda \), ensuring the diagonal of \( D \) consists entirely of the eigenvalue \( \lambda \).
04
Matrix Calculation for A
Substitute \( D = \lambda I \) into the diagonalization formula, giving \( A = PDP^{-1} = P(\lambda I)P^{-1} = \lambda PIP^{-1} = \lambda PP^{-1} = \lambda I \). Consequently, the matrix \( A \) is equal to the scalar matrix \( \lambda I \).
05
Conclusion of Proof
We have shown that under the condition of only one eigenvalue \( \lambda \), \( A = \lambda I \), where \( I \) is the identity matrix. Therefore, \( A \) is in the form of a scalar matrix \( \lambda L \), specifically \( \lambda I \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Eigenvalues
Eigenvalues represent important characteristics of matrices; they play a vital role in determining the properties of a matrix. Essentially, eigenvalues tell us how a linear transformation affects vectors.
An eigenvalue \( \lambda \) of a matrix \( A \) is a scalar such that when it's multiplied by an eigenvector, it doesn't change the direction of the eigenvector. The equation to find eigenvalues is:\[ Av = \lambda v \]
This means multiplying the matrix \( A \) by an eigenvector \( v \) is equivalent to scaling \( v \) by the scalar eigenvalue \( \lambda \).
An eigenvalue \( \lambda \) of a matrix \( A \) is a scalar such that when it's multiplied by an eigenvector, it doesn't change the direction of the eigenvector. The equation to find eigenvalues is:\[ Av = \lambda v \]
This means multiplying the matrix \( A \) by an eigenvector \( v \) is equivalent to scaling \( v \) by the scalar eigenvalue \( \lambda \).
- Eigenvalues help in understanding various physical applications, such as vibrations in mechanical systems.
- They are critical in fields like stability analysis and quantum mechanics.
Eigenvectors
Eigenvectors are vectors that, when a linear transformation is applied, only change by a scalar factor, not in direction.
Each eigenvalue has corresponding eigenvectors, and these are essential in various mathematical and physical contexts. They satisfy the equation \( Av = \lambda v \).
Here are some important features of eigenvectors:
Each eigenvalue has corresponding eigenvectors, and these are essential in various mathematical and physical contexts. They satisfy the equation \( Av = \lambda v \).
Here are some important features of eigenvectors:
- They align with the direction of stretching or compressing in transformations.
- Any non-zero vector parallel to an eigenvector is itself an eigenvector, demonstrating how they represent directions rather than points.
Scalar Matrix
A scalar matrix is a matrix in which every element of the main diagonal is the same scalar, with all off-diagonal elements being zero.
This kind of matrix can be represented as \( \lambda I \), where \( \lambda \) is the scalar, and \( I \) is the identity matrix.
This kind of matrix can be represented as \( \lambda I \), where \( \lambda \) is the scalar, and \( I \) is the identity matrix.
- Such matrices simplify multiplication since each element multiplies only by the scalar value.
- This characteristic makes scalar matrices very useful in linear algebra for scaling operations.
- In our exercise, if a diagonalizable matrix has only one eigenvalue, it becomes a scalar matrix.
Identity Matrix
The identity matrix \( I \) is a special square matrix with ones on the diagonal and zeros elsewhere.
It's like "1" in matrix algebra because multiplying any square matrix by \( I \) leaves it unchanged.
Crucial properties include:
It's like "1" in matrix algebra because multiplying any square matrix by \( I \) leaves it unchanged.
Crucial properties include:
- The identity matrix serves as the multiplicative identity in linear algebra, meaning that for any square matrix \( A \), \( AI = IA = A \).
- It is fundamental in matrix inversion, where \( PP^{-1} = I \) is used.
- In the context of diagonalizing matrices, multiplying by \( I \) does not alter the matrix, which underpins diagonalization processes.
