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Diagonalization Calculator

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Example
Created on 2024-06-20Asked by Samuel Davis (Solvelet student)
Determine if the matrix A=[3113] A = \begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix} is diagonalizable.

Solution

Step 1: Identify the matrix A=[31 13] A = \begin{bmatrix} 3 & 1 \ 1 & 3 \end{bmatrix} . Step 2: Find the eigenvalues λ \lambda by solving the characteristic equation det(A−λI)=0 \text{det}(A - \lambda I) = 0 : det([3−λ1 13−λ])=(3−λ)2−1=λ2−6λ+8=0. \text{det}\left(\begin{bmatrix} 3-\lambda & 1 \ 1 & 3-\lambda \end{bmatrix}\right) = (3-\lambda)^2 - 1 = \lambda^2 - 6\lambda + 8 = 0. Step 3: Solve the quadratic equation λ2−6λ+8=0 \lambda^2 - 6\lambda + 8 = 0 to find the eigenvalues: (λ−4)(λ−2)=0  ⟹  λ1=4,λ2=2. (\lambda - 4)(\lambda - 2) = 0 \implies \lambda_1 = 4, \quad \lambda_2 = 2. Step 4: Determine if A A is diagonalizable by checking if there are enough linearly independent eigenvectors. Since A A has two distinct eigenvalues, it is diagonalizable. Step 5: Conclusion. The matrix A=[31 13] A = \begin{bmatrix} 3 & 1 \ 1 & 3 \end{bmatrix} is diagonalizable. Solved on Solvelet with Basic AI Model
Some of the related questions asked by William Anderson on Solvelet
1. Diagonalize the matrix A=[3−245] A = \begin{bmatrix} 3 & -2 \\ 4 & 5 \end{bmatrix} 2. Use diagonalization to compute An A^n for the matrix A=[3−245] A = \begin{bmatrix} 3 & -2 \\ 4 & 5 \end{bmatrix} and a positive integer n n .,
DefinitionDiagonalization:(Linear algebra) The process of transforming a square matrix into a diagonal matrix by finding a set of eigenvectors with the property that they are linearly independent and they span the entire vector space, and then expressing the matrix as a product of its eigenvectors and a corresponding matrix of eigenvalues. Note: Diagonalization is only valid for diagonalizable matrices, which have a full set of base of linearly independent eigenvectors. This is what is used in linear algebra, differential equations, physics to ease doing matrix math, analysing dynamical systems, solving systems of linear equations and more. For instance: every symmetric matrix can be diagonalized: its eigenvectors and eigenvalues can be computed and the diagonal matrix is then simply constructed using these eigenvectors as columns.
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