Math Problem Statement
Findtheeigenvaluesandeigenvectorsof: A= 3 0 0 0 5 0 0 0 7
Solution
The given matrix is:
3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 7 \end{pmatrix}$$ This is a diagonal matrix, where the diagonal entries are $$3$$, $$5$$, and $$7$$. ### Eigenvalues: For a diagonal matrix, the eigenvalues are simply the diagonal elements. Hence, the eigenvalues of matrix $$A$$ are: $$\lambda_1 = 3, \quad \lambda_2 = 5, \quad \lambda_3 = 7$$ ### Eigenvectors: For a diagonal matrix, the eigenvectors correspond to the standard basis vectors. The eigenvector associated with each eigenvalue is a vector that has a 1 in the position corresponding to the eigenvalue and 0 in all other positions. Thus, the eigenvectors for matrix $$A$$ are: - For $$\lambda_1 = 3$$, the eigenvector is: $$\mathbf{v}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$$ - For $$\lambda_2 = 5$$, the eigenvector is: $$\mathbf{v}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$$ - For $$\lambda_3 = 7$$, the eigenvector is: $$\mathbf{v}_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$$ ### Summary: - Eigenvalues: $$3, 5, 7$$ - Eigenvectors: $$\mathbf{v}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad \mathbf{v}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \quad \mathbf{v}_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$$ Would you like more details or further explanations? Here are 5 related questions: 1. What is the geometric interpretation of eigenvalues and eigenvectors? 2. How would the process change for a non-diagonal matrix? 3. How do you verify that a vector is an eigenvector? 4. What does it mean for a matrix to be diagonalizable? 5. How are eigenvalues used in solving systems of differential equations? **Tip:** For diagonal matrices, the eigenvalues are always the diagonal elements, making the calculation straightforward.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Eigenvectors
Diagonal Matrices
Formulas
Eigenvalue Equation: A * v = λ * v
Theorems
Diagonal Matrix Theorem: For diagonal matrices, the eigenvalues are the diagonal entries
Suitable Grade Level
College Level (Undergraduate)
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