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Schur Decomposition Calculator

Compute A = QUQ⁻¹ with unitary Q and upper triangular U

Enter 2x2 Matrix

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.

Applications

Eigenvalues Stability Matrix Functions Control Theory Quantum Mechanics

Frequently Asked Questions

What is Schur decomposition?▼

A = QUQ⁻¹ where Q is unitary (Q⁻¹=Q*) and U is upper triangular with eigenvalues on diagonal.

Schur vs Jordan?▼

Schur is always possible (unitary Q), Jordan may not be. Both show eigenvalues on diagonal.

How to compute Schur?▼

QR algorithm iteratively transforms matrix to upper triangular form with orthogonal matrices.

Applications of Schur?▼

Eigenvalue computation, stability analysis, matrix functions, control theory, quantum mechanics.

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