# Symmetric Matrix & Skew Symmetric Matrix

Symmetrical matrices and obliquely symmetric matrices are both square matrices. But the difference between them is, a symmetric matrix is equal to its transpose while an obliquely symmetric matrix is a matrix with its negative displacement. If A is a symmetric matrix, then A = AT and if A is a symmetric matrix, then AT = – A. Read: What is an obliquely symmetric matrix?**Also, read: **

- Upper triangular matrix
- Diagonal Matrix
- Unit Matrix

## Symmetrical Matrix

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To understand whether a matrix is symmetric or not, it is very important to know about the matrix’s transpose and how to find it. If we swap the rows and columns of the m × n matrix to get an n × m matrix, the new matrix is called the transpose of the given matrix. There are two possibilities for the number of rows (m) and number of columns (n) of a given matrix:

- If m = n, the matrix is square
- If m n, the matrix is rectangular

For the second case, the transpose of a matrix can never be equal to it. This is because, to be equal, the order of the matrices must be the same. Therefore, the only case where the transpose of a matrix can be equal to it, is when the matrix is square. But this is only the first condition. Even if the matrix is square, its transpose may or may not be equal to it. Example: If (A = begin {bmatrix} 1 & 2cr 3 & 4 end {bmatrix}), then (A’ = begin {bmatrix} 1 & 3cr 2 & 4 end {bmatrix}). Here, we can see that A ≠ A’. Let’s take another example. (B = begin {bmatrix} 1 & 2 & 17cr 2 & 5 & -11 cr 17 & -11 & 9 end {bmatrix}) If we transpose of this matrix we get 🙁 B ‘= begin {bmatrix} 1 & 2 & 17cr 2 & 5 & -11 cr 17 & -11 & 9 end {bmatrix}) We see that B = B’. Whenever this happens with any matrix, i.e. whenever the transpose of a matrix is equal to it, the matrix is called **symmetric matrix**. But how can we find out if a matrix is symmetric or not without finding its transpose? We know that: Read more: What do cemetery keepers sell in pubsIf A = ( [a_{ij}]_ {m × n}) then A ‘= ( [a_{ij}]_ {n × m}) (for all values of i and j) So if for matrix A, (a_ {ij}) = (a_ {ji}) (for all values of i and j) and m = n, then its displacement is equal to itself. Hence the symmetric matrix will always be square. Some examples of symmetric matrices are 🙁 P = begin {bmatrix} 15 & 1cr 1 & -3 end {bmatrix}) (Q = begin {bmatrix} -101 & 12 & 57cr 12 & 1001 & 23 cr 57 & 23 & -10001 ends {bmatrix})

## Properties of Symmetric Matrix

- The addition and difference of two symmetric matrices produces a symmetric matrix.
- If A and B are symmetric matrices and they obey the commutative property, i.e. AB = BA, then the product of A and B is symmetric.
- If the matrix A is symmetric, then An is also symmetric, where n is an integer.
- If A is a symmetric matrix, then A-1 is also symmetric.

## Skew . Symmetric Matrix

A matrix can only be obliquely symmetric if it is square. If the transpose of the matrix is equal to its own negative number, then the matrix is said to be **oblique symmetry**. This means that for matrices that are obliquely symmetric, A’ = – AAlso, for matrices, (a_ {ji}) = – (a_ {ij}) (for all values of i and j) ). The diagonal elements of an obliquely symmetric matrix are zero. This can be proved in the following way: Diagonal elements are characterized by the general formula, (a_ {ij}), where i = j If i = j, then (a_ {ij}) = ( a_ {ii}) = (a_ {jj}) If A is oblique, thenaji = – ajiRead more: Which English word has three consecutive double letters⇒ aii = -aii⇒ 2.aii = 0⇒ aii = 0 So aij = 0, when i = j (for all values of i and j) Some examples of oblique symmetric matrices are 🙁 P = begin {bmatrix} 0 & -5cr 5 & 0 end {bmatrix}) (Q = begin {bmatrix} 0 & 2 & -7cr -2 & 0 & 3 cr 7 & -3 & 0 end {bmatrix})

### Properties of Skew . Symmetric Matrix

- When we add two obliquely symmetric matrices, the resulting matrix is also obliquely symmetric.
- The dot product of an obliquely symmetric matrix is also an obliquely symmetric matrix.
- The diagonal of an obliquely symmetric matrix consists of zero elements, and therefore the sum of the elements in the main diagonals is zero.
- When the identity matrix is added to the obliquely symmetric matrix, the resulting matrix is invertible.
- The determinant of an obliquely symmetric matrix is non-negative

### Determinants of Skew’s Symmetric Matrix

If A is an obliquely symmetric matrix, also a square matrix, then the determinant of A must satisfy the following condition: Det(AT) = det(-A) = (-1) n det(A) Inverse of An obliquely symmetric matrix does not exist because its determinant has an odd degree of 0 and is therefore singular.

### Eigen Value of Skew . Symmetric Matrix

If A is a true obliquely symmetric matrix, then its eigenvalue will be zero. In addition, we can say, that the nonzero eigenvalues of A are not real. Each square matrix can be represented as the sum of a single symmetric matrix and a single, obliquely symmetric matrix. Learn different concepts in math and science by visiting our website BYJU’S.Read more: Calum hood Username & Snapchat | Top Q&A

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