Problem 26
Question
Determine all eigenvalues and corresponding eigenvectors of the given matrix. $$\left[\begin{array}{rrr}1 & 0 & 0 \\\0 & 0 & 1 \\\0 & -1 & 0\end{array}\right]$$.
Step-by-Step Solution
Verified Answer
The eigenvalues of the given matrix are \(\lambda_1 = 1\) and \(\lambda_2 = -1\). Their corresponding eigenvectors are \(v_{\lambda_1} = \left[\begin{array}{c}1 \\ 0 \\ 0\end{array}\right]\) and \(v_{\lambda_2} = \left[\begin{array}{c}0 \\ 1 \\ 1\end{array}\right]\).
1Step 1: Calculate the characteristic determinant
We have to calculate the determinant of the matrix after subtracting \(\lambda\) times the identity matrix. The matrix becomes:
\[
\left[\begin{array}{ccc}
1-\lambda & 0 & 0 \\
0 & -\lambda & 1 \\
0 & -1 & -\lambda
\end{array}\right]
\]
The determinant of this matrix is:
\[
\Delta = (1-\lambda)((-\lambda)(-\lambda) - 1(-1))
\]
2Step 2: Solve the characteristic equation
To find the eigenvalues, we need to solve the characteristic equation, \(\Delta = 0\):
\[
(1-\lambda)(\lambda^2 - 1) = 0
\]
This equation can be factored further as:
\[
(1-\lambda)(\lambda - 1)(\lambda + 1) = 0
\]
Now, we have three possible values for \(\lambda\):
\[
\lambda_1 = 1,
\lambda_2 = -1,
\lambda_3 = 1
\]
3Step 3: Find the corresponding eigenvectors
Now we will plug each eigenvalue into the eigenvector equation to find the corresponding eigenvectors:
For \(\lambda_1 = 1\):
\[
\left[\begin{array}{ccc}
1-1 & 0 & 0 \\
0 & -1 & 1 \\
0 & -1 & -1
\end{array}\right]
\left[\begin{array}{c}
x \\
y \\
z
\end{array}\right] =
\left[\begin{array}{c}
0 \\
0 \\
0
\end{array}\right]
\]
Solving this system, we get the eigenvector for \(\lambda_1\):
\[v_{\lambda_1} =
\left[\begin{array}{c}
1 \\
0 \\
0
\end{array}\right]
\]
For \(\lambda_2 = -1\):
\[
\left[\begin{array}{ccc}
1+1 & 0 & 0 \\
0 & 1 & 1 \\
0 & -1 & 1
\end{array}\right]
\left[\begin{array}{c}
x \\
y \\
z
\end{array}\right] =
\left[\begin{array}{c}
0 \\
0 \\
0
\end{array}\right]
\]
Solving this system, we get the eigenvector for \(\lambda_2\):
\[v_{\lambda_2} =
\left[\begin{array}{c}
0 \\
1 \\
1
\end{array}\right]
\]
Since \(\lambda_3\) is the same as \(\lambda_1\), we don't need to solve the system again as they share the same eigenvector. So, the given matrix has two distinct eigenvalues with corresponding eigenvectors: \(\lambda_1 = 1\) with eigenvector \(v_{\lambda_1}\) and \(\lambda_2 = -1\) with eigenvector \(v_{\lambda_2}\).
Key Concepts
Characteristic EquationDeterminant CalculationMatrix Algebra
Characteristic Equation
When solving for eigenvalues of a matrix, the characteristic equation plays a crucial role. To derive this equation, you first adjust your matrix by subtracting the scalar \( \lambda \) times the identity matrix, transforming your original matrix. This new matrix is known as \( A - \lambda I \). Understanding the transformation of your matrix is essential here, as it sets the groundwork for further calculations.
By calculating the determinant of \( A - \lambda I \), we'll end up with a polynomial equation in terms of \( \lambda \). This polynomial is what we refer to as the characteristic equation of the matrix. Solving this equation yields the eigenvalues—key values that give us information about the original matrix. In essence, the characteristic equation represents the condition a matrix must satisfy to have non-trivial solutions for its eigenvectors.
By calculating the determinant of \( A - \lambda I \), we'll end up with a polynomial equation in terms of \( \lambda \). This polynomial is what we refer to as the characteristic equation of the matrix. Solving this equation yields the eigenvalues—key values that give us information about the original matrix. In essence, the characteristic equation represents the condition a matrix must satisfy to have non-trivial solutions for its eigenvectors.
Determinant Calculation
The determinant calculation is a fundamental process in matrix algebra used to solve the characteristic equation. For our purposes, you calculate the determinant of the matrix \( A - \lambda I \). By expanding the determinant, you effectively unravel the complexity of the matrix into a simpler polynomial relation. This simplifies the search for eigenvalues.
The determinant reveals volumes of information about the matrix, like its invertibility, stability, and potential eigenvalues. These properties are crucial in fields ranging from engineering to computer graphics, where matrix calculations play a key part. Calculate each minor step-by-step and combine them to find the determinant. The key idea is to practice solving for determinants to build familiarity, which is ultimately helpful in understanding the comprehensive picture of eigenvalues.
The determinant reveals volumes of information about the matrix, like its invertibility, stability, and potential eigenvalues. These properties are crucial in fields ranging from engineering to computer graphics, where matrix calculations play a key part. Calculate each minor step-by-step and combine them to find the determinant. The key idea is to practice solving for determinants to build familiarity, which is ultimately helpful in understanding the comprehensive picture of eigenvalues.
Matrix Algebra
Matrix algebra is the comprehensive field in which eigenvalues and eigenvectors reside. This branch of mathematics deals with rectangular arrangements of numbers and offers tools for solving linear systems, among other applications. Matrix algebra introduces various operations, such as addition, multiplication, and transformation, which are the building blocks needed to work with matrices.
Understanding how these transformations affect a matrix helps us understand eigenvalues and eigenvectors better. Eigenvectors provide direction, while eigenvalues provide magnitude, in some sense explaining how the vector space is transformed under the given matrix. In linear transformations, the change made by the matrix can be expressed in terms of its eigenvalues and eigenvectors, providing an elegant way to decompose and understand complex transformations.
Understanding how these transformations affect a matrix helps us understand eigenvalues and eigenvectors better. Eigenvectors provide direction, while eigenvalues provide magnitude, in some sense explaining how the vector space is transformed under the given matrix. In linear transformations, the change made by the matrix can be expressed in terms of its eigenvalues and eigenvectors, providing an elegant way to decompose and understand complex transformations.
- Operations in matrix algebra are foundational. They allow us to analyze the behavior of linear mappings.
- Matrix algebra is integral in modern mathematics, physics, and computer science.
- Understanding these principles can simplify otherwise complicated systems and problems.
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Problem 26
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