Math Problem Statement
Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix. Start 3 By 3 Table 1st Row 1st Column negative 3 2nd Column 9 3rd Column negative 5 2nd Row 1st Column negative 6 2nd Column 10 3rd Column negative 3 3rd Row 1st Column negative 6 2nd Column 6 3rd Column 1 EndTable ; lambdaequals1,3,4
Solution
To diagonalize the given matrix , we first write it out based on the provided entries:
-3 & 9 & -5 \\ -6 & 10 & -3 \\ -6 & 6 & 1 \end{pmatrix}$$ The eigenvalues given are $$\lambda_1 = 1$$, $$\lambda_2 = 3$$, and $$\lambda_3 = 4$$. ### Step 1: Find Eigenvectors For each eigenvalue, we will find the corresponding eigenvector by solving the equation $$(A - \lambda I) \mathbf{v} = 0$$. 1. **For $$\lambda_1 = 1$$**: $$A - I = \begin{pmatrix} -4 & 9 & -5 \\ -6 & 9 & -3 \\ -6 & 6 & 0 \end{pmatrix}$$ We row-reduce the augmented matrix to find the null space. 2. **For $$\lambda_2 = 3$$**: $$A - 3I = \begin{pmatrix} -6 & 9 & -5 \\ -6 & 7 & -3 \\ -6 & 6 & -2 \end{pmatrix}$$ Row-reduce this matrix as well. 3. **For $$\lambda_3 = 4$$**: $$A - 4I = \begin{pmatrix} -7 & 9 & -5 \\ -6 & 6 & -7 \\ -6 & 6 & -3 \end{pmatrix}$$ Again, row-reduce. ### Step 2: Calculate Eigenvectors Performing the row reduction for each $$A - \lambda I$$ gives us the corresponding eigenvectors. ### Step 3: Form the Matrix of Eigenvectors and Diagonal Matrix Let’s denote the eigenvectors found as $$\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$$. The matrix of eigenvectors $$P$$ and the diagonal matrix $$D$$ will be: $$P = \begin{pmatrix} | & | & | \\ \mathbf{v}_1 & \mathbf{v}_2 & \mathbf{v}_3 \\ | & | & | \end{pmatrix}, \quad D = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{pmatrix}$$ ### Step 4: Verify $$A = PDP^{-1}$$ To confirm that $$P$$ diagonalizes $$A$$, check if $$A$$ can be expressed as $$PDP^{-1}$$. ### Conclusion Once you calculate the eigenvectors and complete the matrices $$P$$ and $$D$$, you will have successfully diagonalized the matrix $$A$$. Would you like to go through the row-reduction calculations for each eigenvalue step in detail? Here are some follow-up questions you might consider: 1. What is the row-reduced form of the matrices for each eigenvalue? 2. How do you determine if the eigenvectors are linearly independent? 3. What is the significance of diagonalizing a matrix? 4. Can you provide a practical application for diagonalization? 5. How can you check if the diagonalization is correct? **Tip:** Always check for algebraic multiplicity of eigenvalues to ensure you find a complete set of eigenvectors.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Diagonalization
Eigenvalues and Eigenvectors
Formulas
A - \lambda I = 0
A = PDP^{-1}
Theorems
Diagonalization theorem
Eigenvalue theorem
Suitable Grade Level
Grades 11-12
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