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Matrix Diagonalization Formula:
where \( D \) is a diagonal matrix containing eigenvalues of \( A \) on its diagonal, and \( P \) is a matrix whose columns are the corresponding eigenvectors.
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Matrix diagonalization is the process of finding a diagonal matrix \( D \) and an invertible matrix \( P \) such that \( A = PDP^{-1} \). The diagonal matrix \( D \) contains the eigenvalues of \( A \), and \( P \) contains the corresponding eigenvectors as its columns.
The diagonalization process involves:
Where:
Explanation: Not all matrices are diagonalizable. A matrix is diagonalizable if and only if it has \( n \) linearly independent eigenvectors, where \( n \) is the size of the matrix.
Details: Diagonalization simplifies matrix operations, makes computing powers of matrices efficient, and reveals important properties about linear transformations.
Tips: Enter your square matrix in the format "a,b;c,d" for a 2x2 matrix. Separate columns with commas and rows with semicolons. The matrix must be square (same number of rows and columns).
Q1: What matrices can be diagonalized?
A: A matrix is diagonalizable if it has \( n \) linearly independent eigenvectors (where \( n \) is the matrix size). Symmetric matrices are always diagonalizable.
Q2: What if my matrix isn't diagonalizable?
A: Some matrices can't be diagonalized but can be put in Jordan normal form. The calculator will indicate if diagonalization isn't possible.
Q3: How are eigenvalues calculated?
A: Eigenvalues are found by solving the characteristic equation \( \det(A - \lambda I) = 0 \).
Q4: What's the difference between eigenvectors and eigenvalues?
A: An eigenvector is a non-zero vector that only scales when a linear transformation is applied, and the eigenvalue is the scaling factor.
Q5: Why is diagonalization useful?
A: It simplifies matrix computations, helps solve systems of differential equations, and reveals important properties of the linear transformation.