Diagonalize a 2x2 matrix.
Understand the Problem
The question is asking how to diagonalize a 2x2 matrix, which involves finding a diagonal matrix that is similar to the given matrix and the corresponding transformation matrix.
Answer
$A = PDP^{-1}$, where $D$ is diagonal and $P$ contains eigenvectors.
Answer for screen readers
The matrix $A$ can be diagonalized as $A = PDP^{-1}$, where $D$ is the diagonal matrix containing the eigenvalues and $P$ is the matrix of eigenvectors.
Steps to Solve
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Identify the matrix Start with the given 2x2 matrix, which we'll denote as $A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$.
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Find the Eigenvalues To find the eigenvalues, solve the characteristic polynomial, given by:
$$ \text{det}(A - \lambda I) = 0 $$
Where $I$ is the identity matrix and $\lambda$ represents the eigenvalues. For a 2x2 matrix, this becomes:
$$ \text{det} \begin{pmatrix} a - \lambda & b \ c & d - \lambda \end{pmatrix} = 0 $$
Calculate the determinant:
$$ (a - \lambda)(d - \lambda) - bc = 0 $$
From which you can find the characteristic equation.
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Solve for Eigenvalues Now, solve the characteristic equation to find the eigenvalues $\lambda_1$ and $\lambda_2$.
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Find the Eigenvectors For each eigenvalue $\lambda_i$, substitute back into the equation:
$$ (A - \lambda_i I)\mathbf{v} = 0 $$
Where $\mathbf{v}$ is the eigenvector. This gives us a system of equations to solve for the eigenvectors.
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Construct the Matrix of Eigenvectors Once you obtain the eigenvectors for both eigenvalues, construct the matrix $P$ where each column corresponds to an eigenvector.
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Form the Diagonal Matrix Create a diagonal matrix $D$ where the diagonal entries are the eigenvalues, such that:
$$ D = \begin{pmatrix} \lambda_1 & 0 \ 0 & \lambda_2 \end{pmatrix} $$
- Verify the Diagonalization Check that $A$ can be expressed as:
$$ A = PDP^{-1} $$
This confirms that $A$ is diagonalized by $P$.
The matrix $A$ can be diagonalized as $A = PDP^{-1}$, where $D$ is the diagonal matrix containing the eigenvalues and $P$ is the matrix of eigenvectors.
More Information
Diagonalization is a fundamental concept in linear algebra, often used in various applications such as solving differential equations and transforming systems of equations into simpler forms. A 2x2 matrix can always be diagonalized unless it has defective eigenvalues (i.e., not enough linearly independent eigenvectors).
Tips
- Failing to determine whether the matrix is diagonalizable. A matrix is not diagonalizable if it does not have enough linearly independent eigenvectors.
- Incorrectly calculating the determinant or solving the characteristic polynomial.
- Confusing the order of eigenvalues and eigenvectors in constructing matrix $P$ and diagonal matrix $D$.
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