Compute A = QUQ⁻¹ with unitary Q and upper triangular U
Enter 2x2 Matrix
Enter 3x3 Matrix
Result
Step-by-Step Derivation
Schur Decomposition Formula
A = Q U Q⁻¹
Q*Q = I (unitary)
U is upper triangular
diag(U) = eigenvalues of A
Schur decomposition states that any square complex matrix is unitarily similar to an upper triangular matrix with eigenvalues on the diagonal. For real matrices, a real Schur form exists with Q orthogonal and U quasi-upper triangular.
⚠Schur form always exists (with complex eigenvalues if needed), while Jordan form may require generalized eigenvectors for defective matrices.
What is Schur Decomposition?
Schur decomposition is a matrix factorization that transforms a square matrix into upper triangular form using a unitary (or orthogonal for real symmetric) similarity transformation. The diagonal of U contains the eigenvalues of A, making Schur decomposition a powerful tool for eigenvalue computation and matrix analysis.
Unitary Matrix
Q satisfies Q⁻¹ = Q* (conjugate transpose). Columns are orthonormal.
Upper Triangular U
U has eigenvalues on diagonal, zeros below. For real matrices, can have 2x2 blocks.
Existence
Always exists over complex numbers. Real Schur over real numbers.
QR Algorithm
Numerical method: iteratively apply QR decomposition to converge to Schur form.
Teaching Example: 2x2 matrix A = [[3,1],[1,3]].
1. Eigenvalues: 3+1=4 and 3-1=2
2. Eigenvectors: [1,1] and [1,-1]
3. Q = [[1/√2,1/√2],[1/√2,-1/√2]] (unitary)
4. U = [[4,0],[0,2]] (upper triangular)
5. Verify: Q U Q⁻¹ = A.
Free online calculators and tools covering mathematics, unit conversion, text processing, and daily life. Accurate, fast, mobile-friendly, and completely free to use.