Skew Symmetric Matrix

In linear algebra, a symmetric matrix is defined by its symmetry along the main diagonal. If a square matrix A = A^T(transpose of A), then the matrix is symmetric. In other words, the value at row i and column j equals the value at row j and column i, i.e., A[i, j] = A [j, i].  

Matrix A is a skew symmetric matrix if A = -A^T, where A^T is the transpose of A. If A = [a_ij]_n is a skew symmetric matrix, then a_ij = -a_ji. It means that all elements present diagonally in a skew-symmetric matrix are zero. 
 

Now, let’s learn how to represent skew symmetric matrices. Let B = [b_ij]_n be an n × n matrix, Matrix B is a skew symmetric b_ij = -b_ji for all 1 ≤ i, j ≤ n. Where n is the natural number and bij is the element at the i-th row and j-th column.      

All the diagonal elements of a skew symmetric matrix are always zero, as b_ii = -b_ii as i = j
Adding b_ii to both sides: b_ii + b_ii = 0 
2b_ii = 0
b_ii = 0
 

Skew symmetric matrices follow certain properties so that the concept can be used in algebra, physics, etc. In this section, we will learn the properties of skew symmetric matrices. 

• The sum of two skew symmetric matrices is skew symmetric. 
  (A + B)^T = A^T + B^T = -A - B = -(A + B), so (A + B)^T = -(A + B)

• The product of a skew symmetric matrix and a scalar ‘k’ results in skew symmetric
  (kA)^T = k(A)^T = k(-A) = -kA, so (kA)^T = -kA

• If A and B are skew symmetric and AB = -BA, then AB is skew symmetric
  (AB)^T = B^TA^T = (-B)(-A) = BA = -AB. Therefore, (AB)^T = -AB

• In all skew symmetric matrices, the diagonal elements are zero. 
  For a skew symmetric A, 
  a_ij = -a_ji
  So, the diagonals will be: 
  a_ii = -a_ii → a_ii + a_ii = 0
  2a_ii = 0 → a_ii = 0

• The sum of diagonal elements (trace) is always zero, as all diagonal elements are zero.
  
   
• For a skew symmetric matrix A over the real numbers, A + I is invertible in many cases, but this requires proof. E.g., for even-order matrices.
  
   
• If a skew symmetric matrix A is invertible, then its inverse is also skew-symmetric. For A to be a skew matrix: A = -A^T
  Taking the inverse on both sides; 
  A^-1 = (-A^-1)^T = -(A^-1)^T
  So, A^-1 = -(A^-1)^T

In this section, we will be discussing the theorems on skew symmetric matrices. These theorems are useful in matrix decomposition, transformations, and applications. 

Theorem 1: For a square matrix A, A - A^T is skew symmetric
For any square matrix A, we should prove that A - A^T is skew symmetric.
We can take help from the properties of skew symmetric matrices, such as: 

• (A + B)^T = A^T + B^T
• (kA)^T = kA^T
• (A^T)^T = A

For A - A^T  to be skew symmetric, let’s prove that B = A - A^T
To prove B = A - A^T, let’s take transpose on both sides.
B^T = (A - A^T)^T
Now, we need to use the transpose of a different property.
B^T  = A^T + (-A^T)^T
B^T = A^T - (A^T)^T
We know that (A^T)^T = A 
Therefore, B^T  = A^T - A
B^T  = -(A - A^T)  B^T = -B
So, -B = -(A - A^T)
We can now say that B = (A - A^T) is skew-symmetric.

Theorem 2: Decomposing a Square Matrix
Like all matrices, square matrices also encode complex relationships. It is important to decompose a square matrix into similar components to understand their properties and structure. A square matrix is expressed as the sum of a symmetric matrix, and a skew-symmetric matrix will be proved in this theorem. This decomposition is significant in applications like physics. 

In this theorem, we will be using the following properties:

• (A^T)^T = A
• (A + B)^T = A^T + B^T
• (kA)^T = kA^T
• If M^T = M, then matrix M is symmetric
• If N^T = -N, then matrix N is skew symmetric

If A is a square matrix, it can be written as:
A = 1/2 (A + A^T) + 1/2 (A - A^T)
Let’s consider, 
P = 1/2 (A + A^T) 
Q = 1/2 (A - A^T)

Taking the transpose of both P and Q
PT = (1/2 (A + A^T))^T
= ½(A^T + (A^T)^T)
= ½(A^T + A)
P = ½(A^T + A)
So, PT = P

QT = (1/2 (A - A^T))^T
= ½(A^T - (A^T)^T)
= ½ (A^T - A)
= ½ (A^T - A)
-Q = ½ (A - A^T)
So, Q^T = - Q
So, the square matrix A can be written as the sum of a symmetric matrix P and a skew symmetric matrix Q. 

Theorem 3: For a skew symmetric matrix A and any matrix B, the matrix BTAB is skew symmetric.

Let’s take the help of the following properties:

• (AB)^T = B^TA^T
• (A^T)^T = A
•  A^T = -A

In this theorem, we shall prove (B^TAB)^T = -B^TAB
Start with:
(B^T AB)^T = (B^T (AB))^T
Now use the transpose of a product rule:
= (AB)^T (B^T)^T
Since (AB)^T = B^T A^T we get:
(B^T AB)^T = B^TA^TB
Since A^T = -A, we can make the substitution.
Therefore, (B^T AB)^T = B^T(-A)B
(B^T AB)^T = -B^T AB

So, if A is a skew symmetric matrix, then B^T AB is a skew symmetric matrix. 
 

According to the theorem of skew symmetric matrices, the sum of any symmetric and skew symmetric matrix results in a square matrix. 

Determinant of Skew Symmetric Matrix
 

For any skew symmetric matrix with an odd order, the determinant is always zero. Let’s verify this with an example:

Finding the determinant of a matrix A
|A| = m₁₁ × Cofactor₁₁ + m₁₂ × cofactor₁₂ + m₁₃ × cofactor₁₃
So, |A| = 0 × Cofactor₁₁ + 2 × cofactor₁₂ + (-4) × cofactor₁₃
|A| = 2 × cofactor₁₂ + (-4) × cofactor₁₃

Finding cofactor C₁₂, eliminating row 1 and column 2, 
So,  -2      3

         4      0

det = (-2)(0) - (4)(3) = 0 - 12 = -12
So, C₁₂ = (-1)^{1 + 2} × (-12) 
= -1 × -12 = 12

Finding cofactor C₁₃, eliminating row 1 and column 3, 
So,    -2     0

          4     -3

det = (-2)(-3) - (4)(0) = 6 - 0 = 6
So, C₁₃ = (-1)^{1 + 3} × (6) 
= 1 × 6 = 6

So, |A| = 2 × cofactor₁₂ + (-4) × cofactor₁₃
= 2 × 12 + (-4) × 6
= 24 + -24 
= 0

So, the determinant of a skew symmetric matrix with an odd order is 0. 
 

For all skew symmetric matrices, the eigenvalue is always zero or imaginary. For a skew symmetric matrix A, the eigenvalue λ of A with a corresponding eigenvector x can be represented as:
Ax = λx

Before we proceed further, let us understand what eigenvalue and eigenvector are. In a matrix, a vector changes direction when a specific matrix acts on it. But an eigenvector doesn’t change direction, it only gets stretched or shrunk. The amount by which it gets stretched or shrunk is known as eigenvalue. 
Multiplying both sides by the conjugate transpose of x, x ^-T:
x ^-T Ax =  x  ^-T =  ||x²||

Here, 
x ^-T Ax is a dot product, so it is commutative, x^T- Ax  = (Ax )^T- x= x^T A^T- x
As A is a skew symmetric, A^T = -A
Substituting A^T = -A: 
x^T A^T x = -x^T(-A)x = -x^T Ax
Taking the conjugate of Ax,
Ax = x
Now, we have
-x^T Ax = -x^T x = - ||x||²
- ||x||² = ||x||²
As ||x||² ≠ 0, we get
 = - 
The value of  is either 0 or an imaginary number.

In real life, we use the skew symmetric matrix in fields like physics, mathematics, computer graphics, etc. In this section, we will learn some real-life applications of skew symmetric matrices.

• In physics, we use skew symmetric matrices to represent angular velocity vectors. This allows rotational motion to be expressed mathematically in a compact and convenient form. 

• In computer graphics and robotics, skew symmetric matrices are used to compute rotations and transformations.

• For 3D image reconstruction in MRI and CT scans, we use skew symmetric matrices to capture the geometry of camera movement.    

• In ocean modeling, skew symmetric matrices play an important role as they are used to describe the rotational flow in fluids. It is used in predicting hurricanes, turbulence in aviation, and ocean currents.