As we are all aware by now that equal matrices have equal dimensions, hence only the square matrices can be symmetric or skew-symmetric form. Now, what is a symmetric matrix and a skew-symmetric matrix? Let’s learn about them in further detail below.
A square matrix A is said to be symmetric if aij = aji for all i and j, where aij is an element present at (i,j)th position (ith row and jth column in matrix A) and aji is an element present at (j,i)th position (jth row and ith column in matrix A). In other words, we can say that matrix A is said to be symmetric if the transpose of matrix A is equal to matrix A itself (AT=A). Let’s take an example of a matrix,
It is symmetric matrix because aij = aji for all i and j. Here, a12 = a21= 3, a13 = a31= 8 and a23 = a32 = -4 In other words, the transpose of Matrix A is equal to Matrix A itself (AT=A) which means matrix A is symmetric.
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Square matrix A is said to be skew-symmetric if aij =−aji for all i and j. In other words, we can say that matrix A is said to be skew-symmetric if transpose of matrix A is equal to negative of matrix A i.e (AT =−A). Note that all the main diagonal elements in the skew-symmetric matrix are zero. Let’s take an example of a matrix
It is skew-symmetric matrix because aij =−aji for all i and j. Here, a12 = -6 and a21= -6 which means a12= −a21. Similarly, this condition holds true for all other values of i and j.
For any square matrix A with real number entries, A + A′ is a symmetric matrix and A – A′ is a skew-symmetric matrix.
Proof: Let B =A+A′, then B′= (A+A′)′
= A′ + (A′)′ (as (A + B)′ = A′ + B′)
= A′ +A (as (A′)′ =A)
= A + A′ (as A + B = B + A) =B
Therefore, B = A+A′is a symmetric matrix
Now let C = A – A′
C’ = (A–A′)′=A′–(A′)′ (Why?)
= A′ – A (Why?)
=– (A – A′) = – C
Hence, A – A′ is a skew-symmetric matrix.
Any Square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix.
Proof: Let A be a square matrix then, we can write A = 1/2 (A + A′) + 1/2 (A − A′). From the Theorem 1, we know that (A + A′) is a symmetric matrix and (A – A′) is a skew-symmetric matrix.
Since for any matrix A, (kA)′ = kA′, it follows that 1 / 2 (A+A′) is a symmetric matrix and 1 / 2 (A − A′) is a skew-symmetric matrix. Thus, any square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix.
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Solved Examples for You
Question 1: If A and B are symmetric matrices, then ABA is
- Skew – Symmetric
Answer : Given A and B are Symmetric Matrices
⇒ AT = A and BT = B
Now, take (ABA)T
(ABA)T = ATBTAT
(ABA)T = ABA
Hence, ABA is also Symmetric
Question 2: Say true or false: If A & B are symmetric matrices of same order then is symmetric.
Answer : Given, A and B are symmetric matrices, therefore we have:
A’ = A and B’ = B……….(i)
Consider, (AB – BA)’ = (AB)’ – (BA)’……………[ Since, (A – B)’ = A’ – B’]
= B’A’ – A’ B’ ……………[ Since, (AB)’ = B’ A’]
= BA – AB …..[by (i)]
= – (AB – BA)
Therefore, (AB – BA)’ = – (AB – BA)
Thus, (AB – BA) is a skew-symmetric matrix
Question 3: Explain a symmetric matrix?
Answer: Symmetric matrix refers to a matrix in which the transpose is equal to itself.
Question 4: Explain a skew symmetric matrix?
Answer: A matrix can be skew symmetric only if it happens to be square. In case the transpose of a matrix happens to be equal to the negative of itself, then one can say that the matrix is skew symmetric. Therefore, for a matrix to be skew symmetric, A’=-A.
Question 5: What is meant by the inverse of a symmetric matrix?
Answer: The inverse of a symmetric matrix happens to be the same as the inverse of any matrix. As such, any matrix, whose multiplication takes place (from the right or the left) with the matrix in question, results in the production of the identity matrix. An important point to understand is that not all symmetric matrices are invertible.
Question 6: Can we say that a positive definite matrix is symmetric?
Answer: A positive definite matrix happens to be a symmetric matrix that has all positive eigenvalues. Note that all the eigenvalues are real because it’s a symmetric matrix all the eigenvalues are real. Consequently, it makes sense to discuss them being positive or negative.