The Schur decomposition guarantees that any complex square matrix can be brought to upper triangular form via a unitary transformation. The Schur decomposition is a fundamental result in linear algebra with several important applications:

• Eigenvalues: The eigenvalues of A are the diagonal entries of the upper triangular matrix U. This makes the Schur decomposition a valuable tool for computing eigenvalues, especially for large matrices.
• Matrix Analysis: Many matrix analysis problems are simplified when dealing with triangular matrices. The Schur decomposition allows us to transform a general matrix into a triangular form, making it easier to analyze its properties.
• Numerical Algorithms: The Schur decomposition is used in various numerical algorithms, including those for solving eigenvalue problems and computing matrix exponentials. It's numerically more stable than directly trying to diagonalize a matrix.
• Theoretical Results: The Schur decomposition is a key ingredient in the proofs of other important theorems in linear algebra.

Example: Schur Decomposition for an Eigenvalue Problem

Given the matrix:

$$ A=\left[\begin{array}{ll} 4 & 2 \\ 1 & 3 \end{array}\right] $$

we want to find a unitary matrix $Q$ such that:

$$ Q^* A Q=U $$

where $U$ is an upper triangular matrix and the eigenvalues of $A$ appear on its diagonal.

Step 1: Compute the Eigenvalues of $A$ Eigenvalues are found by solving:

$$ \begin{aligned} & \operatorname{det}(A-\lambda I)=0 \\ & \left|\begin{array}{cc} 4-\lambda & 2 \\ 1 & 3-\lambda \end{array}\right|=0 \end{aligned} $$

Expanding the determinant:

$$ \begin{gathered} (4-\lambda)(3-\lambda)-(2 \cdot 1)=0 \\ 12-4 \lambda-3 \lambda+\lambda^2-2=0 \\ \lambda^2-7 \lambda+10=0 \end{gathered} $$

Factoring:

$$ (\lambda-5)(\lambda-2)=0 $$

Thus, the eigenvalues are:

$$ \lambda_1=5, \quad \lambda_2=2 $$

Step 2: Compute Eigenvectors of $A$ For $\lambda_1=5$, solve: