Symmetric and Skew Symmetric Matrices

Here you will learn what are symmetric and skew symmetric matrices with examples.

Let’s begin –

Symmetric and Skew Symmetric Matrices

Symmetric Matrix

A square matrix A = \([a_{ij}]\) is called a symmetric matrix, if

\(A^T\) = A or \(a_{ij}\) = \(a_{ji}\) for all i, j. 

for example, the matrix A = \(\begin{bmatrix} 3 & -1 & 1 \\ -1 &  2 & 5 \\ 1 & 5 &  -2  \end{bmatrix}\) is symmetric, because

\(a_{12}\) = -1 = \(a_{21}\), \(a_{13}\) = 1 = \(a_{31}\), \(a_{23}\) = 5 = \(a_{32}\) i.e. \(a_{ij}\) = \(a_{ji}\) for all i, j.

It follows from the definition of a symmetric matrix that A is symmetric, iff

\(a_{ij}\) = \(a_{ji}\)   for all i, j.

\(\iff\)  \(A_{ij}\) = \((A^T)_{ij}\)  for all i, j

\(\iff\)  A = \(A^T\)

Thus, a square matrix A is a symmetric matrix iff \(A^T\) = A.

Matrices A = \(\begin{bmatrix} a & h & g \\ h &  b & f \\ g & f &  c  \end{bmatrix}\), B = \(\begin{bmatrix} 2 + i & 1 & 3 \\ 1 &  2 & 3 + 2i \\ 3 & 3 + 2i &  4  \end{bmatrix}\) are symmetric matrices because \(A^T\) = A and \(B^T\) = B.

Skew-Symmetric Matrix

A square matrix A = \([a_{ij}]\) is called a skew-symmetric matrix, if

\(A^T\) = -A or \(a_{ij}\) = -\(a_{ji}\) for all i, j. 

for example, the matrix A = \(\begin{bmatrix} 0 & 2 & -3 \\ -2 &  0 & 5 \\ 3 & -5 &  0  \end{bmatrix}\) is skew symmetric, because

\(a_{12}\) = 2, \(a_{21}\) = -2 \(\implies\) \(a_{12}\) = -\(a_{21}\) \(a_{13}\) = -3, \(a_{31}\) = 3 \(\implies\) \(a_{13}\) = -\(a_{31}\) 

and,  \(a_{23}\) = 5,  \(a_{32}\) = – 5 \(\implies\) \(a_{23}\) = -\(a_{32}\)

It follows from the definition of a skew symmetric matrix that A is skew symmetric, iff

\(a_{ij}\) = -\(a_{ji}\)   for all i, j.

\(\iff\)  \(A_{ij}\) = -\((A^T)_{ij}\)  for all i, j

\(\iff\)  A = -\(A^T\)

Thus, a square matrix A is a skew symmetric matrix iff \(A^T\) = -A.

Matrices A = \(\begin{bmatrix} 0 & 2i & 3 \\ -2i &  0 & 4 \\ -3 & -4 & 0  \end{bmatrix}\), B = \(\begin{bmatrix} 0 & -3 & 5 \\ 3 &  0 & 2 \\ -5 & -2 &  0  \end{bmatrix}\) are skew symmetric matrices because \(A^T\) = -A and \(B^T\) = -B.

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