Symmetric Matrix

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are used to represent linear transformations, perform matrix operations, and solve systems of linear equations. A matrix with ‘m’ rows and ‘n’ columns is denoted as m × n. A square matrix has equal numbers of rows and columns, n × n.
 

A symmetric matrix satisfies the condition A = A^T, i.e., the matrix and its transpose are equal. This suggests that it must be a square matrix, and each element in the (i, j) position must equal the elements in the position (j, i). i.e., a_ij = a_ji.
           
This is a symmetric matrix.
 

Skew-symmetry is a property of square matrices. It is different from a symmetric matrix in the following ways:
 

Some properties that help identify symmetric matrices are listed below:

• The sum of two symmetric matrices is also symmetric.

• Eigenvalues of a real symmetric matrix are real.

• Every diagonal matrix is symmetric.

• The product of two symmetric matrices may or may not be symmetric.

• The transpose of a symmetric matrix is always equal to the original matrix (A^T = A).

• They can be diagonalized using orthogonal matrices, meaning there exists an orthogonal matrix P such that P^TAP =  D, where D is a diagonal matrix.
   

Let us understand the two important theorems and their proofs for symmetric matrices.

Theorem 1: 

For any square matrix B with real number entries: 
B + B^T is a symmetric matrix, and
B - B^T is a skew-symmetric matrix.
Proof:
Let us take A = B + B^T
Taking the transpose of A,
A^T = (B + B^T)^T = B^T + (B^T)^T = B^T + B = B + B^T = A
Since A^T = T, this confirms that B + B^T is symmetric.
Now, let’s take C = B - B^T
Taking the transpose of C,
C^T = (B - B^T)^T = B^T - (B^T)^T = B^T - B = - (B - B^T) = - C
Since C^T = - C, this proves that B - BT is skew-symmetric.
Let us take an example:

Step 1: Compute its transpose

Step 2: Compute B + B^T

The matrix is symmetric because (B + B^T)^T = B + B^T

Step 3: Compute B - B^T

This matrix is skew-symmetric because (B - B^T)^T = - (B - B^T)

Theorem 2: Any square matrix can be written as the sum of a symmetric matrix and a skew-symmetric matrix.
Proof:
Let B be a square matrix.

We use the following identity: B = 12(B +B^T) + 12(B-B^T) 
Where,
B^T is the transpose of matrix B
12(B +B^T) is symmetric because: 12(B+B^T)^T=12(B^T+(B^T)^T) =12(B^T+B) = 12(B + B^T)
12(B -B^T) is skew-symmetric because: 12(B-B^T)^T=12(B^T-B) = -12(B - B^T)
Hence, the square matrix B can be expressed as the sum of a symmetric and skew-symmetric matrix.
Let us use the same matrix from the previous example:

Step 1: Calculate the symmetric part:

Step 2: Calculate the skew-symmetric part:

Step 3:     Verify their sum

From analyzing mechanical stress in bridges to simplifying data in machine learning, symmetric matrices help model, optimize, and solve real-life problems efficiently. Some real-life applications of symmetric matrices are listed below:

• Representing stiffness matrix in structural engineering:
  In civil and mechanical engineering, symmetric matrices represent the stiffness matrices in finite element methodology. These matrices help simulate how structures like bridges or buildings may deform under force.

• Scaling, shearing, and reflective transformations in computer graphics:
  Symmetric matrices help define how objects appear on-screen when they are being moved, rotated, or mirrored.

• Intermediate calculations in Google PageRank algorithm:
  The Google PageRank matrix isn't symmetric itself but requires intermediate calculations involving symmetric matrices. They simplify eigenvalue problems or balance link weights. 

• Blurring and filtering in image processing:
  In digital image processing, symmetric matrices are used for tasks like blurring, edge detection and noise reduction. Symmetry in kernel filters ensures uniform behavior in all directions.

• Network analysis:
  In graph theory, for example social networks, transportation systems, or communication systems, adjacency matrices of undirected graphs are symmetric.