Symmetric and Skew Symmetric Matrices: Definition, Properties, Proofs

Matrix: The development of the concept of matrices results from an attempt to obtain compact and straightforward methods of solving a system of linear equations. Matrices are not only used in solving systems of linear equations. They are also used in electronic spreadsheet programs in computers, which are used in cost estimation of products, budgeting, analysing results of an experiment.

Matrices also have a wide range of probability, engineering, genetics, and statistics applications. Matrices can also represent even rotation or reflection through a plane. There are different types of matrices. In this article, let us learn about symmetric and skew symmetric matrices. In this article, we will provide detailed information on Symmetric and Skew Symmetric Matrices.

Transpose of a Matrix

Transpose of a matrix \(A\) is obtained by interchanging its row and column into column and row. Transpose of a matrix \(A\) is denoted by \(\mathrm{A}^{\mathrm{T}}\) Where \(T\) is used to denote the transpose.

Note: If two matrices are related as \(A^{\mathrm{T}}=\mathrm{B}\) then each of them is known as the transpose of each other.

Properties of Transpose

1. For any matrix, \(\left(A^{T}\right)^{T}=A\)

2. For any matrix \(A\) and scalar \(k,(\mathrm{kA})^{\mathrm{T}}=k A^{\mathrm{T}}\)

3. \((A \pm B)^{\mathrm{T}}=A^{\mathrm{T}} \pm \mathrm{B}^{\mathrm{T}}\), where \(A\) and \(B\) are two comparable matrices. This is true for \(n\) comparable matrices.

4. \(\operatorname{Tr}\left(A^{T}\right)=\operatorname{Tr}(A)\) where \(\operatorname{Tr}(A)\) denotes the trace of a matrix \(A\).

5. \((A B)^{T}=B^{T} A^{T}\) where \(A\) and \(B\) are conformable matrices for the product, this property is true for \(n\) matrices, provided the matrices are conformable for the product.

Definition of Symmetric Matrix

Any square matrix equal to its transpose matrix is a symmetric matrix. It is denoted as

\(A=A^{T}\)

where, \(\mathrm{A}^{\mathrm{T}}\) denotes the transpose of a matrix \(A\).

Condition for a Symmetric Matrix

For any matrix to be a symmetric matrix, the matrix must be a square matrix, because only the dimensions of a square matrix are equal to the dimension of its transpose.

• \([\mathrm{A}]_{\mathrm{m} \times \mathrm{m}}\) is a symmetric matrix if and only if \(a_{i j}=a_{j i}\), such that for all \(i,j,i,j \le m\), where \(a_{i j}\) are the elements of the matrix.
• \((i, j)^{t h}\) element of matrix \(A\) and \((j, i)^{t h}\) element of matrix \(\mathrm{A}^{\mathrm{T}}\) are equal.

Properties of Symmetric Matrix

1. If \(A\) is a symmetric matrix, then \(\mathrm{A}=\mathrm{A}^{\mathrm{T}}\)
2. If \(A\) is a symmetric matrix, then \(kA\) is also a symmetric matrix, where \(k\) is scalar.
3. The product of the matrix and its transpose is always symmetric, i.e \(\mathrm{AA}^{\mathrm{T}}=\left(\mathrm{AA}^{\mathrm{T}}\right)^{\mathrm{T}}\)
4. The sum of two symmetric matrices is also symmetric.
5. If \(A\) is a symmetric matrix, then \(\mathrm{A}^{\mathrm{n}}\) is also symmetric (Where \(\mathrm{n} \in \mathrm{N}\)).
6. If \(A\) and \(B\) are the symmetric matrices of the same order, then
   i) \(AB\) is symmetric iff \(AB=BA\)
   ii) \(AB+BA\) is symmetric
7. The maximum number of distinct elements in a symmetric matrix of order \(n\) is \(\frac{\mathrm{n}(\mathrm{n}+1)}{2}\)
8. If \(A\) and \(B\) are the square matrix of the same order and \(A\) is a symmetric matrix, then \(\mathrm{B}^{\mathrm{T}} \mathrm{AB}\) is also symmetric.

Definition of Skew Symmetric Matrix

Any square matrix equal to the negative of its transpose matrix is known as a skew symmetric matrix. It is denoted as

\(\mathrm{A}=-\mathrm{A}^{\mathrm{T}}\)

where \(A^{T}\) denotes the transpose of a matrix \(A\)

• For a matrix to be a skew symmetric matrix, the first condition is the matrix should be a square matrix because only the dimension of a square matrix is equal to the dimension of its transpose matrix.
• \({{\rm{[A]}}_{{\rm{m \times m}}}}\) matrix is a skew symmetric matrix if and only if \(a_{i j}=-a_{j i}\) such that for all \(i,j,i,j \le m\) where \(a_{i j}\) are the elements of matrix.
• \({(i,j)^{th}}\) element of matrix \(A\) and \((j, i)^{t h}\) element of matrix \(A^{T}\) are equal in magnitude and opposite in sign.

Properties of Skew Symmetric Matrix

1. In a skew symmetric matrix, all the leading diagonal elements are zero.

For skew symmetric matrix \(a_{i j}=-a_{j i}\) for all \(i,j \le n\) where \(n\) is the order of a matrix.

For diagonal elements \(i=j\)

Therefore, \(a_{i i}=-a_{i i} \Rightarrow 2 a_{i i}=0\) and \(a_{i i}=0\) for all \(i \leq n\)

2. If \(A\) is a skew symmetric matrix, then \(kA\) is also a skew symmetric matrix, where \(k\) is scalar.

3. The sum of two skew symmetric matrices is also skew symmetric.

4. If \(A\) is a skew symmetric matrix, then \(A^{2 n+1}\) is also skew symmetric (Where \(\mathrm{n} \in \mathrm{N}\) ).

5. If \(A\) is a skew symmetric matrix, then \(\mathrm{A}^{2 \mathrm{n}}\) is not a skew symmetric (Where \(\mathrm{n} \in \mathrm{N}\) ).

6. If \(A\) and \(B\) are skew symmetric matrices of the same order, then \(AB-BA\) is skew symmetric.

7. If \(A\) and \(B\) are the square matrices of the same order and \(A\) is a skew symmetric matrix, then \(\mathrm{B}^{\mathrm{T}} \mathrm{AB}\) is also skew symmetric.

8. If \(A\) is a skew symmetric matrix and \(C\) is a non-zero column matrix , then \(C^{\mathrm{T}} \mathrm{AC}\) is a null matrix.

Proof of Property Theorems

1. A matrix that is both symmetric and skew symmetric is a null matrix.

2. For any square matrix \(A, \mathrm{~A}+\mathrm{A}^{\mathrm{T}}\) is symmetrc and \(\mathrm{A}-\mathrm{A}^{\mathrm{T}}\) is a skew symmetric matrix.

Proof:
Given \(A\) is any square matrix.

Let, \(\mathrm{B}=\mathrm{A}+\mathrm{A}^{\mathrm{T}}\) …… \((1)\)

\(\mathrm{C}=\mathrm{A}-\mathrm{A}^{\mathrm{T}}\) ……. \((2)\)

Now taking transpose on both sides of \((1\)) and \((2\)) we get,

\(B^{T}=\left(A+A^{T}\right)^{T}\) and \(C^{\mathrm{T}}=\left(\mathrm{A}-\mathrm{A}^{\mathrm{T}}\right)^{\mathrm{T}}\)

\(\mathrm{B}^{\mathrm{T}}=\mathrm{A}+\mathrm{A}^{\mathrm{T}}=\mathrm{B}\) and \(C^{\mathrm{T}}=A^{\mathrm{T}}-\mathrm{A}=-\left(\mathrm{A}-\mathrm{A}^{\mathrm{T}}\right)=-\mathrm{C}\)

Therefore, \(\mathrm{B}^{\mathrm{T}}=\mathrm{B}\) and \({{\mathop{\rm C}\nolimits} ^T} = – C\)

Hence, proved that \(B=A+A^{T}\) is a symmetric matrix and \(C=A-A^{T}\) is a skew symmetric matrix.

3. Every square matrix \(A\) can be uniquely expressed as a sum of a symmetric and a skew symmetric matrix.

\(A=\left(\frac{A+A^{T}}{2}\right)+\left(\frac{A-A^{T}}{2}\right)\)

where,

\(\frac{A+A^{T}}{2}\) is a symmetric matrix

\(\frac{\mathrm{A}-\mathrm{A}^{\mathrm{T}}}{2}\) is a skew symmetric matrix

Proof:

From the above property, for any square matrix \(A, \mathrm{~A}+\mathrm{A}^{\mathrm{T}}\) is a symmetric matrix and \(\mathrm{A}-\mathrm{A}^{\mathrm{T}}\)  is a skew symmetric matrix.

Therefore, \(\frac{A+A^{T}}{2}\) is a symmetric matrix and \(\frac{A-A^{T}}{2}\) is a skew symmetric matrix.

Let, \(B=\frac{A+A^{T}}{2}\) and \(\mathrm{C}=\frac{\mathrm{A}-\mathrm{A}^{\mathrm{T}}}{2}\)

Now,

\(A=\frac{A+A^{T}+A-A^{T}}{2}\)

\(=\frac{A+A^{T}}{2}+\frac{A-A^{T}}{2}\)

\(A=\mathrm{B}+\mathrm{C}\)

Therefore, \(A=B+C=\) symmetric matrix \(+\) Skew symmetric matrix.

Hence, every square matrix \(A\) can be uniquely expressed as a sum of a symmetric and a skew symmetric matrix.

4. Product of skew symmetric and a symmetric matrices is skew symmetric, when the product is commutative.

Determinant of Skew Symmetric Matrix

If \(A\) is a skew symmetric matrix of odd order, then \(\operatorname{det}(A)=0\) and for even order \(\operatorname{det}(A)\) is a non-zero perfect square.

Solved Examples – Symmetric and Skew Symmetric Matrices

Q.1. If \(X = \left[ {\begin{array}{*{20}{c}} 0&{2y}& -2 \\ 3&1&3 \\ {3x}&3&{ – 1} \end{array}} \right]\) is a symmetric matrix, then find the value of \(x\) and \(y\).
Sol:
Given: \(X = \left[ {\begin{array}{*{20}{c}} 0&{2y}& -2 \\ 3&1&3 \\ {3x}&3&{ – 1} \end{array}} \right]\) is a symmetric matrix.
\(\therefore \left[ {\begin{array}{*{20}{c}} 0&{2y}& -2 \\ 3&1&3 \\ {3x}&3&{ – 1} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0&3&{3x} \\ {2y}&1&3 \\ { – 2}&3&{ – 1} \end{array}} \right]\) \([\because\) For a symmetric matrix, \(\left.X=X^{T}\right]\)
Since the corresponding elements of two equal matrices are equal, we get
\(2 y=3\) and \(3 x=-2\)
\(\therefore y=\frac{3}{2}\) and \(x=\frac{-2}{3}\)

Q.2. If \(A = \left[ {\begin{array}{*{20}{c}} {2x + 3}&4&5 \\ { – 4}&0&{ – 6} \\ { – 5}&6&{ – 2x – 3} \end{array}} \right]\) is a skew symmetric matrix, then find the value of \(x\)
Sol:
Given: \(A = \left[ {\begin{array}{*{20}{c}} {2x + 3}&4&5 \\ { – 4}&0&{ – 6} \\ { – 5}&6&{ – 2x – 3} \end{array}} \right]\) is a skew symmetric matrix.
We know that if a matrix is skew symmetric, then its transpose is equal to negative of the matrix.
\(A^{T}=-A\)
\( \Rightarrow \left[ {\begin{array}{*{20}{c}} {2x + 3}&{ – 4}&{ – 5} \\ 4&0&{ 6} \\ 5&{ -6}&{ – 2x – 3} \end{array}} \right] = \, – \left[ {\begin{array}{*{20}{c}} {2x + 3}&4&5 \\ { – 4}&0&{ – 6} \\ { – 5}&-6&{ – 2x – 3} \end{array}} \right]\)
\( \Rightarrow \left[ {\begin{array}{*{20}{c}} {2x + 3}&{ – 4}&{ – 5} \\ 4&0&6 \\ 5&{ – 6}&{ – 2x – 3} \end{array}} \right] = \,\left[ {\begin{array}{*{20}{c}} { – 2x – 3}&{ – 4}&{ – 5} \\ 4&0&6 \\ 5&{ – 6}&{2x + 3} \end{array}} \right]\)
Since the corresponding elements of two equal matrices are equal, by equating the corresponding elements, we get,
\(2 x+3=-2 x-3\)
\(\Rightarrow 4 x=-6\)
\(\therefore x=\frac{-3}{2}\)
Hence, for \(x=\frac{-3}{2}\) the matrix is skew symmetric matrix.

Q.3. Express the matrix \(\left[ {\begin{array}{*{20}{c}} 6&5 \\ 2&4 \end{array}} \right]\) as a sum of a symmetric and a skew symmetric matrix.
Sol:
Given: \(X = \left[ {\begin{array}{*{20}{c}} 6&5 \\ 2&4 \end{array}} \right]\)
We know that \(X=\frac{1}{2}\left(X+X^{T}\right)+\frac{1}{2}\left(X-X^{T}\right)\), where \(\frac{1}{2}\left(X+X^{T}\right)\) is symmetric and \(\frac{1}{2}\left(X-X^{T}\right)\) is skew symmetric matrix.
Now, \(X + {X^T} = \left[ {\begin{array}{*{20}{c}} 6&5 \\ 2&4 \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 6&2 \\ 5&4 \end{array}} \right]\)
\(X + {X^T} = \left[ {\begin{array}{*{20}{c}} {12}&7 \\ 7&8 \end{array}} \right]\)
\( \Rightarrow \frac{1}{2}\left( {X + {X^T}} \right) = \left[ {\begin{array}{*{20}{c}} 6&{\frac{7}{2}} \\ {\frac{7}{2}}&4 \end{array}} \right]\)
\(\therefore\) Symmetric part of \(X = \frac{1}{2}\left( {X + {X^T}} \right) = \left[ {\begin{array}{*{20}{c}} 6&{\frac{7}{2}} \\ {\frac{7}{2}}&4 \end{array}} \right]\)
Now, \(X – {X^T} = \left[ {\begin{array}{*{20}{c}} 6&5 \\ 2&4 \end{array}} \right] – \left[ {\begin{array}{*{20}{c}} 6&2 \\ 5&4 \end{array}} \right]\)
\(X – {X^T} = \left[ {\begin{array}{*{20}{c}} 0&3 \\ { – 3}&0 \end{array}} \right]\)
\(\therefore \frac{1}{2}\left( {X – {X^T}} \right) = \left[ {\begin{array}{*{20}{c}} 0&{\frac{3}{2}} \\ { – \frac{3}{2}}&0 \end{array}} \right]\)
\(\therefore\) Skew Symmetric part of \(X = \frac{1}{2}\left( {X – {X^T}} \right) = \left[ {\begin{array}{*{20}{c}} 0&{\frac{3}{2}} \\ { – \frac{3}{2}}&0 \end{array}} \right]\)
\(\therefore X = \frac{1}{2}\left( {X + {X^T}} \right) + \frac{1}{2}\left( {X – {X^T}} \right) = \left[ {\begin{array}{*{20}{c}} 6&{\frac{7}{2}} \\ {\frac{7}{2}}&4 \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 0&{\frac{3}{2}} \\ { – \frac{3}{2}}&0 \end{array}} \right]\)

Q.4. Write a \(4 \times 4\) matrix which is both symmetric and skew symmetric matrix.
Sol:
We have to find a \(4 \times 4\) matrix which is both symmetric and skew symmetric.
We know that a square matrix equal to its transpose is known as a symmetric matrix, and a square matrix equal to its negative of transpose is known as a symmetric matrix.
Let the matrix be \(x\)
For symmetric matrix \(X=X^{T}\)
\(X-X^{T}=0\) (i)
For skew symmetric matrix \(X=-X^{T}\)
\(X+X^{T}=0\) (ii)
Adding (i) and (ii), we get
\(2 X=0\)
\(X=0\)
\(\therefore X = \left[ {\begin{array}{*{20}{c}} 0&0&0&0 \\ 0&0&0&0 \\ 0&0&0&0 \\ 0&0&0&0 \end{array}} \right]\)
Hence, null matrix is the required \(4 \times 4\) square matrix that is both symmetric and skew symmetric.

Q.5. If  \(\left[ {\begin{array}{*{20}{c}} 0&7&9 \\ { – 7}&0&{ – 6} \\ { – 9}&6&0 \end{array}} \right]\) is a skew symmetric matrix, then show that \({\left[ {\begin{array}{*{20}{c}} 0&7&9 \\ { – 7}&0&{ – 6} \\ { – 9}&6&0 \end{array}} \right]^8}\) is a symmetric matrix.
Sol:
Given: \(\left[ {\begin{array}{*{20}{c}} 0&7&9 \\ { – 7}&0&{ – 6} \\ { – 9}&6&0 \end{array}} \right]\)  is a symmetric matrix.
To show \({\left[ {\begin{array}{*{20}{c}} 0&7&9 \\ { – 7}&0&{ – 6} \\ { – 9}&6&0 \end{array}} \right]^8}\) is a symmetric matrix.
We know that if \(A\) is a skew symmetric matrix. Then,
\(A^{T}=-A\).
Now, \(A^{2 n}=(A \cdot A \cdot A \ldots(2 n\text { times }))\) (Where \(n \in N\) )
\(\Rightarrow\left(A^{2 n}\right)^{T}=(A \cdot A \cdot A \ldots(2 n \text { times }))^{T}\)
\(\Rightarrow\left(A^{2 n}\right)^{T}=\left(A^{T} \cdot A^{T} \cdot A^{T} \ldots(2 n\right.\) times \(\left.)\right)\)
\(\Rightarrow\left(A^{2 n}\right)^{T}=((-A) \cdot(-A) \cdot(-A) \ldots(2 n\text { times }))\)
\(\Rightarrow\left(A^{2 n}\right)^{T}=A^{2 n}\)
So, the even power of a skew symmetric matrix is symmetric matrix.
Hence, \({\left[ {\begin{array}{*{20}{c}} 0&7&9 \\ { – 7}&0&{ – 6} \\ { – 9}&6&0 \end{array}} \right]^8}\) is a symmetric matrix.

Summary

The arrangement of elements in \(m\) rows and \(n\) columns to form a rectangular array is known as a matrix. Matrix with dimension \(\mathrm{m} \times \mathrm{n}\) is known as rectangular matrix. If \(m=n\), then the matrix is called a square matrix. The transpose of any matrix \(A\) is obtained by interchanging its rows and columns. Any square matrix equal to its transpose matrix is a symmetric matrix i.e., \({a_{ij}} = {a_{ji}}\).

Any square matrix equal to the negative of its transport matrix is known as a skew symmetric matrix i.e., \({a_{ij}} = \,-{a_{ji}}\). Every square matrix A can be uniquely expressed as a sum of a symmetric and skew symmetric matrices. For a skew symmetric of odd order, \(\operatorname{det}(A)=0\) and for even order \(\operatorname{det}(A)\) is a non-zero perfect square.

Frequently Asked Questions (FAQs)

Q.1. How do you find symmetric and skew symmetric matrices?
Ans: Any square matrix \(A\) is symmetric, if \(\mathrm{A}=\mathrm{A}^{\mathrm{T}}\), and its is skew symmetric, if \(\mathrm{A}=-\mathrm{A}^{\mathrm{T}}\)

Q.2. What are symmetric and skew symmetric matrices?
Ans: Any square matrix \(A\) is a symmetric matrix if \(\mathrm{A}=\mathrm{A}^{\mathrm{T}}\) and skew symmetric matrix if \({\rm{A}} = – {{\rm{A}}^{\rm{T}}}\). To find the symmetric or skew symmetric matrix, first, find the transpose of the matrix if transpose is equal to the given matrix, then the matrix is symmetric otherwise, check for skew symmetric.

Q.3. What is a skew symmetric matrix with an example?
Ans: Any square matrix equal to the negative of its transpose matrix is known as a skew symmetric matrix. It is denoted as \(\mathrm{A}=-\mathrm{A}^{\mathrm{T}}\). Where \(A^{T}\)denotes the transpose of a matrix \(A\). i.e \(a_{i j}=-a_{j i}\) , such that for all \(i.j\) where \(i, j \leq m\) where \(a_{i j}\) are the elements of the matrix.
Example: Let \(A = \left[ {\begin{array}{*{20}{c}} 0&7&9 \\ { – 7}&0&{ – 6} \\ { – 9}&6&0 \end{array}} \right]\) then
\({A^T} = \left[ {\begin{array}{*{20}{c}} 0&{ – 7}&{ – 9} \\ 7&0&6 \\ 9&{ – 6}&0 \end{array}} \right] = \, – A\)
Since, \(A^{T}=-A\) or \(\mathrm{A}=-\mathrm{A}^{\mathrm{T}}\)
Therefore, \(A\) is a skew symmetric matrix.

Q.4. Can a matrix be both symmetric and skew symmetric?
Ans: Yes, a null matrix is both symmetric as well as skew symmetric.

Q.5. Is null matrix symmetric or skew symmetric?
Ans: Null matrix is both symmetric and skew symmetric matrix, as it holds both the properties: For any null matrix \(A, A=A^{T}\) and \(A = \, – {A^T}\).

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