An involutory matrix is its own inverse. Applying the transformation twice returns to the original state. This property makes involutory matrices useful in applications like reflections and toggles.
⚠Every involutory matrix is diagonalizable with eigenvalues ±1. The trace equals the number of +1 eigenvalues minus the number of -1 eigenvalues.
What is an Involutory Matrix?
An involutory matrix is a square matrix that is its own inverse, meaning A² = I where I is the identity matrix. This implies A⁻¹ = A. Involutory matrices represent operations that are their own inverses, like reflections across axes or swapping operations.
Self-Inverse
A⁻¹ = A, applying twice returns to original.
Determinant
det(A) = ±1, product of eigenvalues.
Eigenvalues
Only possible eigenvalues are +1 and -1.
Diagonalizable
Always diagonalizable with orthogonal eigenvectors.
Example: R = [[0,1],[1,0]] (swap matrix)
1. R² = [[0,1],[1,0]] · [[0,1],[1,0]] = [[1,0],[0,1]] = I ✓
2. R is involutory ✓
3. R swaps rows, swapping twice returns to original
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