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The Symmetric determinant in linear algebra is one of the matrices widely used in machine learning. The Symmetric determinant is usually taken as a square matrix, and it is always equivalent to its transpose. AT is the transpose of a matrix A.
Determinant of a Matrix
Determinant of the matrix in algebra means the function of the symmetrical or square matrix entries. To get an idea, let’s find out the determinant for the 2 x 2 Symmetric determinant,
A= a11 a12 a21 a22
Then, A= a11 a12 a21 a22 = a11 x a22 – a21 x a22
Symmetric determinant definition
The determinant is said to be a symmetric determinant when it remains the same even after taking its transpose. It is used in algebra, similar to the square matrix. In a determinant, the numbers are arranged in a row and a column to form an array which is in a rectangular or square shape, eg
1. 2 x 2 symmetric determinant 1 3 4 5
2. 3 x 3 symmetric determinant
Determinant Transpose
A determinant transpose is defined as the interchanged rows and columns of the original determinant. In this, the first row changes its position with the second column, and the second row changes its position with the first column, and this procedure goes on.
If we take A as the determinant, where
A= 1 2 2 3
Thus, its transpose AT will be,
AT= 1 2 2 3
Thus, here you can see the transpose of the determinant and the determinant both remain the same. Thus, it is a Symmetric determinant.
Properties of a Symmetric Determinant
A Symmetric determinant will be obtained if the sum and difference of the two symmetric determinants are taken. For example, if A and B are the symmetric determinants, the A-B and A+B will be symmetric.
If A and B commutates, then and then only AB=BA. Where A and B are the Symmetric determinant and AB is their product.
If there is A, which is a symmetric determinant, and an integer n, then An will also be a symmetric determinant.
If A is a symmetric determinant, then A’s inverse will also be a symmetric determinant.
The product of the matrix and the transpose of the Symmetric determinant is always symmetric. That is, A.AT will be symmetric.
The determinant will be symmetric only and only when the determinant is of a square matrix.
If the Symmetric determinant has a scalar multiple, then the product of both the determinants will be symmetric.
2 Theorems of the Symmetric Determinant
A symmetric determinant has two main theorems. Let’s see what they are, along with their proof.
Theorem 1:
Take A = C + CT
Transpose of A will be, AT = (C + CT )T = C T + (C T )T = C T + C = C + C T = A
C + CTcan be said as a Symmetric determinant
Next, let B= C – CT
BT = ( C + ( – CT ))T = CT + ( – CT )T = CT – ( CT )T = CT– C = – ( C – CT ) = – B
B − BTcan be said as a skew-symmetric determinant
Thus, the theorem states that if the square matrix C has real number elements, then the sum of the matrix and its transpose (C + CT) and difference (C – CT) will be Symmetric determinant and skew-symmetric determinant, respectively.
Theorem 2:
Let’s take A as a square matrix. Then,
A = (1/2) × (A + AT) + (1/2 ) × (A – AT).
Here, ATis the transpose of the square matrix A.
- If A + ATis a Symmetric determinant, then (1/2) × (A + AT) is also a Symmetric determinant.
- If A – AT is a skew-symmetric matrix, then (1/2 ) × (A – AT) is also a skew-determinant matrix.
Thus, the theorem states that the sum of a skew-Symmetric matrix and a Symmetric determinant is a square matrix.
Conclusion
Thus, for the matrix to be the Symmetric determinant, the matrix has to be symmetrical. The sum and difference of the two symmetric determinants are symmetric. The product of the matrix and the transpose of the Symmetric determinant is always symmetric. If the Symmetric determinant has a scalar multiple, then the product of both the matrices will be symmetric. Determinant of the matrix in algebra means the function of the symmetrical or square matrix entries. For a symmetric determinant, the transpose of the matrix and the matrix both remain the same.
