Symmetric and Skew Symmetric Matrix

A matrix is a collection of numbers represented in the form of rows and columns. In this article, we will learn about symmetric and skew-symmetric matrices, their definition, related formulas, and the difference between the two matrices.

Symmetric Matrix:

A matrix is said to be a symmetric matrix if the transpose of the matrix is equal to the original matrix. Transpose of a matrix: If in a matrix we interchange the positions of the items in the rows and columns and change the row into a column and vice versa, the resultant matrix is known as the transpose of the original matrix. It is denoted by a single inverted comma. The transpose of a rectangular matrix can never be equal to the original matrix. For example, if we have a matrix of the order 2 x 3, then the transpose will be of the order 3 x 2, which creates an entirely different matrix. Hence a rectangular matrix can never be symmetrical. However, a square matrix with the order m x m will have a transpose of the same order. So, the foremost condition to find whether a matrix is symmetric is that it should be a square matrix. The second condition for a symmetric matrix is that its transpose should yield a matrix identical to the primary matrix.

Mathematically we can find a symmetric matrix using the following concept: If K = [k ab ] r×c then K’ = [k ab ] c×r (where r = rows and c = columns). This implies that for all the values of ‘a’ and ‘b’, the transpose K’ is equal to the matrix K. Let us now discuss some properties of the symmetric matrix:

• A symmetric matrix results after the summation or subtraction of two symmetric matrices.
• Symmetric matrices obey commutative law, which states that if there are two matrices K and L, then the product of these two matrices is also symmetric, KL = LK.
• The matrix K n is symmetric for a symmetric matrix K, and similarly, K-1 is also symmetric.

Skew Symmetric Matrix

Now that we have discussed the symmetric matrix, let us discuss the converse of a symmetric matrix, i.e., skew-symmetric matrix. Just like a symmetric matrix, a skew-symmetric matrix must have a square primary matrix. A matrix is said to be skew-symmetric if the transpose of any matrix yields another matrix that is negative of the original matrix. Mathematically, K’ = -K or k ab = -k ba is a skew- symmetric matrix. The elements present in the diagonal of a skew-symmetric matrix are always equivalent to zero. Let us prove the above statement. We know that diagonal elements are always 11, 22, 33, 44, and so on. This implies a = b. Therefore, in a skew-symmetric matrix

k_ab = – k_ba

⇒ k_aa = – k_aa

⇒ 2.k_aa = 0

⇒ k_aa = 0

So, k ab = 0, for all the diagonal elements of a skew-symmetric matrix

Properties of Skew Symmetric Matrix

• The addition and multiplication of skew-symmetric matrices result in the formation of another skew-symmetric matrix.
• An invertible matrix is obtained by adding an identity matrix to a skew-symmetric one.
• The skew symmetric matrix consists of a positive determinant.