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


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})

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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
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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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