【Advanced】Complex Matrix Operations: Schur Decomposition and Jordan Form in MATLAB

# Advanced Topic: Advanced Matrix Operations - Schur Decomposition and Jordan Form in MATLAB

## 2.1 Basic Concepts and Properties of Schur Decomposition

Schur decomposition is a matrix factorization technique that decomposes a matrix into the product of an upper triangular matrix and a unitary matrix.

**Basic Concepts:**

Given a complex matrix A, its Schur decomposition is represented as:

A = Q * T * Q^H

Where:

* Q is a unitary matrix, meaning Q^H * Q = I
* T is an upper triangular matrix, whose diagonal elements are the eigenvalues of A

**Properties:**

* Schur decomposition is unique; that is, for a given A, there exists a unique Q and T that satisfy the decomposition.
* The diagonal elements of T are the eigenvalues of A, arranged according to their algebraic multiplicity.
* The columns of Q are the eigenvectors of A, i.e., A * Q = Q * T.

## 2. Schur Decomposition Theory and Algorithms

### 2.1 Basic Concepts and Properties of Schur Decomposition

**Schur Decomposition** is a matrix factorization technique that decomposes a square matrix into the product of an upper triangular matrix and a unitary matrix. For a real symmetric matrix, its Schur decomposition form is:

A = Q * T * Q^T

Where:

***A** is the real symmetric matrix to be decomposed
***Q** is a unitary matrix, meaning its transpose is equal to its inverse
***T** is an upper triangular matrix, whose diagonal elements are the eigenvalues of A

Schur decomposition has the following properties:

***Orthogonality:** Q is a unitary matrix, thus Q^T * Q = I, where I is the identity matrix. This indicates that the columns of Q are orthogonal.
***Diagonalization:** T is a diagonal matrix, hence its diagonal elements are the eigenvalues of A.
***Uniqueness:** For a given A, its Schur decomposition is unique, except that the order of the columns of the unitary matrix Q may vary.

### 2.2 Algorithm Implementation of Schur Decomposition

#### 2.2.1 QR Algorithm

The QR algorithm is an iterative algorithm used to compute the Schur decomposition of a matrix. The algorithm decomposes A into the product of an upper triangular matrix and a unitary matrix through a series of QR decompositions.

**Algorithm Steps:**

1. Initialize A as A0.
2. For k = 1, 2, ..., n-1, perform the following steps:
* Perform a QR decomposition on A_k to obtain A_k = Q_k * R_k.
* Update A_k+1 = R_k * Q_k.
3. Let T = A_n, Q = Q_1 * Q_2 * ... * Q_n.

**Code Block:**

def schur_qr(A):
     """
     Computes the Schur decomposition of matrix A using the QR algorithm.

     Parameters:
         A: The real symmetric matrix to be decomposed

     Returns:
         T: An upper triangular matrix whose diagonal elements are the eigenvalues of A
         Q: A unitary matrix whose columns are the eigenvectors of A
     """
     n = A.shape[0]
     Q = np.eye(n)
     for k in range(n-1):
         A, Q_k = qr(A)
         Q = Q @ Q_k
     T = A
     return T, Q

**Logical Analysis:**

The code block implements the QR algorithm. It first initializes A as A0, then iteratively updates A through a series of QR decompositions. In each iteration, it decomposes A into the product of an upper triangular matrix and a unitary matrix and updates A and Q. Finally, it returns the upper triangular matrix T and the unitary matrix Q.

#### 2.2.2 Divide and Conquer Method

The divide and conquer method is a recursive algorithm used to compute the Schur decomposition of a matrix. The algorithm breaks down A into smaller blocks and recursively computes the Schur decomposition of these blocks.

**Algorithm Steps:**

1. If A is a 2x2 matrix, directly compute its eigenvalues and eigenvectors.
2. Otherwise, decompose A into the following form:

A = [A11 A12]
     [A21 A22]

***
***
***bine the Schur decompositions of A11 and A22 to obtain the Schur decomposition of A.

**Code Block:**

def schur_divide(A):
     """
     Computes the Schur decomposition of matrix A using the divide and conquer method.

     Parameters:
         A: The real symmetric matrix to be decomposed

     Returns:
         T: An upper triangular matrix whose diagonal elements are the eigenvalues of A
         Q: A unitary matrix whose columns are the eigenvectors of A
     """
     n = A.shape[0]
     if n == 2:
         return schur_2x2(A)
     else:
         # Decompose A
         A11 = A[:n//2, :n//2]
         A12 = A[:n//2, n//2:]
         A21 = A[n//2:, :n//2]
         A22 = A[n//2:, n//2:]

         # Recursively compute the Schur decomposition of the submatrices
         T11, Q1 = schur_divide(A11)
         T22, Q2 = schur_divide(A22)

         # Combine the Schur decompositions of the submatrices
         T = np.block([[T11, A12 @ Q2],
                      [Q1.T @ A21, T22]])
         Q = np.block([[Q1, np.zeros((n//2, n//2))],
                      [np.zeros((n//2, n//2)), Q2]])
         return T, Q

**Logical Analysis:**

The code block implements the divide and conquer method. It first checks if A is a 2x2 matrix; if so, it directly