Transpose and Symmetric Matrices

Transpose of a Matrix

The transpose of a matrix is obtained by changing its rows into columns and its columns into rows. दूसरे शब्दों में, matrix की rows और columns की positions आपस में interchange हो जाती हैं। The transpose of a matrix $A$ is denoted by $A^T$.

If

$$A=\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}$$

then its transpose is

$$A^T=\begin{pmatrix}1&4\\2&5\\3&6\end{pmatrix}$$

यहाँ $A$ की पहली row $\begin{pmatrix}1&2&3\end{pmatrix}$, $A^T$ की पहली column बन गई है और दूसरी row दूसरी column बन गई है।

Order of the Transpose

If a matrix $A$ is of order $m\times n$, then its transpose $A^T$ is of order $n\times m$.

$$A_{m\times n}\quad\Longrightarrow\quad A^T_{n\times m}$$

अर्थात transpose लेने पर rows और columns की संख्या आपस में बदल जाती है। For example, a matrix of order $2\times3$ will have a transpose of order $3\times2$.

General Form of Transpose

If

$$A=[a_{ij}]_{m\times n}$$

then

$$A^T=[a_{ji}]_{n\times m}$$

यहाँ $a_{ij}$ में row और column का index interchange होकर $a_{ji}$ हो जाता है। इसलिए transpose में element की position बदल जाती है।

Transpose of a Square Matrix

Consider the square matrix:

$$A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$$

Its transpose is:

$$A^T=\begin{pmatrix}a&c\\b&d\end{pmatrix}$$

Principal diagonal के elements $a$ और $d$ अपनी position पर ही रहते हैं, जबकि diagonal के opposite side के elements $b$ और $c$ आपस में interchange हो जाते हैं।

Properties of Transpose

1. Transpose of Transpose

If we take the transpose of a matrix twice, we get the original matrix back.

$$(A^T)^T=A$$

पहली बार transpose लेने पर rows और columns interchange होते हैं। दूसरी बार transpose लेने पर वे फिर अपनी original position में आ जाते हैं।

2. Transpose of a Sum

The transpose of the sum of two matrices is equal to the sum of their transposes.

$$(A+B)^T=A^T+B^T$$

इसका अर्थ है कि पहले matrices को add करके transpose लें या दोनों matrices का अलग-अलग transpose लेकर add करें, result समान होगा।

3. Transpose of a Difference

The transpose of the difference of two matrices is:

$$(A-B)^T=A^T-B^T$$

इसलिए subtraction के साथ भी transpose इसी प्रकार distribute होता है।

4. Transpose of a Scalar Multiple

If $k$ is a scalar, then:

$$(kA)^T=kA^T$$

Scalar $k$ transpose लेने के बाद भी बाहर ही रहता है।

5. Transpose of a Product

The transpose of the product of two matrices is obtained by reversing their order.

$$(AB)^T=B^TA^T$$

यहाँ सबसे important बात यह है कि multiplication का order reverse हो जाता है। पहले $A$ फिर $B$ था, लेकिन transpose के बाद $B^T$ पहले और $A^T$ बाद में आता है।

Example: Transpose of a Sum

Let

$$A=\begin{pmatrix}1&2\\3&4\end{pmatrix}, \qquad B=\begin{pmatrix}5&6\\7&8\end{pmatrix}$$

First calculate $A+B$:

$$A+B= \begin{pmatrix}6&8\\10&12\end{pmatrix}$$

Therefore:

$$(A+B)^T= \begin{pmatrix}6&10\\8&12\end{pmatrix}$$

Now calculate the individual transposes:

$$A^T=\begin{pmatrix}1&3\\2&4\end{pmatrix}, \qquad B^T=\begin{pmatrix}5&7\\6&8\end{pmatrix}$$

Then:

$$A^T+B^T= \begin{pmatrix}6&10\\8&12\end{pmatrix}$$

Hence:

$$(A+B)^T=A^T+B^T$$

इस example से यह property directly verify हो जाती है।

Symmetric Matrix

A square matrix $A$ is called a symmetric matrix if its transpose is equal to the matrix itself.

$$A^T=A$$

इसका मतलब है कि matrix को transpose करने पर कोई change नहीं होता। Matrix के principal diagonal के दोनों sides में corresponding elements equal होते हैं।

Example of a Symmetric Matrix

Consider:

$$A=\begin{pmatrix}2&5\\5&7\end{pmatrix}$$

Its transpose is:

$$A^T=\begin{pmatrix}2&5\\5&7\end{pmatrix}$$

Therefore:

$$A^T=A$$

Hence, $A$ is a symmetric matrix. यहाँ diagonal के ऊपर और नीचे corresponding elements $5$ और $5$ equal हैं।

General Form of a Symmetric Matrix

A general symmetric matrix of order $2\times2$ can be written as:

$$A=\begin{pmatrix}a&b\\b&d\end{pmatrix}$$

Notice that the elements above and below the principal diagonal are equal. यानी $a_{12}=a_{21}$.

A general symmetric matrix of order $3\times3$ is:

$$A=\begin{pmatrix} a&b&c\\ b&d&e\\ c&e&f \end{pmatrix}$$

Here:

$$a_{12}=a_{21},\qquad a_{13}=a_{31},\qquad a_{23}=a_{32}$$

यानी principal diagonal के दोनों तरफ समान दूरी पर स्थित corresponding elements equal होते हैं।

Condition for a Symmetric Matrix

For a matrix $A=[a_{ij}]$ to be symmetric:

$$a_{ij}=a_{ji}$$

for all valid values of $i$ and $j$.

इस condition का मतलब है कि किसी भी element की position $(i,j)$ हो, तो उसके mirror position $(j,i)$ पर वही element होना चाहिए।

Skew-Symmetric Matrix

A square matrix $A$ is called a skew-symmetric matrix if:

$$A^T=-A$$

इसका मतलब है कि transpose लेने पर matrix का प्रत्येक element अपने negative के बराबर हो जाता है। Principal diagonal के बाहर corresponding elements एक-दूसरे के negatives होते हैं।

Example of a Skew-Symmetric Matrix

Consider:

$$A=\begin{pmatrix}0&4\\-4&0\end{pmatrix}$$

Its transpose is:

$$A^T=\begin{pmatrix}0&-4\\4&0\end{pmatrix}$$

Also:

$$-A=\begin{pmatrix}0&-4\\4&0\end{pmatrix}$$

Therefore:

$$A^T=-A$$

Hence, $A$ is a skew-symmetric matrix. यहाँ $4$ के opposite position पर $-4$ है, यानी दोनों elements एक-दूसरे के negatives हैं।

Diagonal Elements of a Skew-Symmetric Matrix

An important property of a skew-symmetric matrix is that all its diagonal elements are zero.

Since

$$A^T=-A$$

for a diagonal element $a_{ii}$, its transpose position remains the same. Therefore:

$$a_{ii}=-a_{ii}$$

Adding $a_{ii}$ to both sides:

$$2a_{ii}=0$$

Hence:

$$a_{ii}=0$$

इसलिए skew-symmetric matrix की principal diagonal पर हर element हमेशा zero होता है।

General Form of a Skew-Symmetric Matrix

A general skew-symmetric matrix of order $2\times2$ is:

$$A=\begin{pmatrix}0&a\\-a&0\end{pmatrix}$$

A general skew-symmetric matrix of order $3\times3$ is:

$$A=\begin{pmatrix} 0&a&b\\ -a&0&c\\ -b&-c&0 \end{pmatrix}$$

ध्यान दें कि diagonal के सभी elements zero हैं और diagonal के opposite elements एक-दूसरे के negatives हैं।

Difference Between Symmetric and Skew-Symmetric Matrices

Property	Symmetric Matrix	Skew-Symmetric Matrix
Basic condition	$A^T=A$	$A^T=-A$
Corresponding elements	$a_{ij}=a_{ji}$	$a_{ij}=-a_{ji}$
Diagonal elements	May be any value	Always zero
Matrix must be	Square	Square

इस difference को याद रखने का आसान तरीका है: symmetric matrix में opposite elements same होते हैं, जबकि skew-symmetric matrix में opposite elements negative होते हैं।

Principal Diagonal

The diagonal extending from the top-left element to the bottom-right element is called the principal diagonal.

For example:

$$A=\begin{pmatrix} \boxed{2}&5&7\\ 5&\boxed{4}&8\\ 7&8&\boxed{9} \end{pmatrix}$$

इस matrix में $2$, $4$ और $9$ principal diagonal के elements हैं। Symmetric matrix में diagonal के दोनों sides के corresponding elements equal होते हैं।

How to Check a Matrix is Symmetric

1. Check that the matrix is square.
2. Find its transpose $A^T$.
3. Compare $A^T$ with $A$.
4. If $A^T=A$, the matrix is symmetric.

यानी symmetric matrix check करने के लिए सबसे direct तरीका है transpose निकालकर original matrix से compare करना।

Example: Checking a Symmetric Matrix

Check whether

$$A=\begin{pmatrix} 1&3&5\\ 3&2&7\\ 5&7&4 \end{pmatrix}$$

is symmetric.

Its transpose is:

$$A^T=\begin{pmatrix} 1&3&5\\ 3&2&7\\ 5&7&4 \end{pmatrix}$$

Since:

$$A^T=A$$

therefore, $A$ is symmetric.

यहाँ $a_{12}=a_{21}=3$, $a_{13}=a_{31}=5$ और $a_{23}=a_{32}=7$ हैं। इसलिए symmetric condition satisfy होती है।

How to Check a Matrix is Skew-Symmetric

1. Check that the matrix is square.
2. Find $A^T$.
3. Find $-A$.
4. Compare $A^T$ and $-A$.
5. If $A^T=-A$, the matrix is skew-symmetric.

इसके साथ diagonal elements भी zero होने चाहिए।

Example: Checking a Skew-Symmetric Matrix

Check whether

$$A=\begin{pmatrix} 0&2&-3\\ -2&0&5\\ 3&-5&0 \end{pmatrix}$$

is skew-symmetric.

Its transpose is:

$$A^T=\begin{pmatrix} 0&-2&3\\ 2&0&-5\\ -3&5&0 \end{pmatrix}$$

Also:

$$-A=\begin{pmatrix} 0&-2&3\\ 2&0&-5\\ -3&5&0 \end{pmatrix}$$

Therefore:

$$A^T=-A$$

Hence, $A$ is skew-symmetric. यहाँ diagonal के सभी elements $0$ हैं और बाकी corresponding elements opposite signs में हैं।

Decomposition of a Square Matrix

Every square matrix $A$ can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix.

$$A=\frac{1}{2}(A+A^T)+\frac{1}{2}(A-A^T)$$

इस expression में पहला part symmetric matrix देता है और दूसरा part skew-symmetric matrix देता है।

The symmetric part is:

$$\frac{1}{2}(A+A^T)$$

and the skew-symmetric part is:

$$\frac{1}{2}(A-A^T)$$

Example of Decomposition

Let

$$A=\begin{pmatrix}2&3\\1&4\end{pmatrix}$$

Then:

$$A^T=\begin{pmatrix}2&1\\3&4\end{pmatrix}$$

The symmetric part is:

$$\frac{1}{2}(A+A^T) = \frac{1}{2} \begin{pmatrix} 4&4\\ 4&8 \end{pmatrix} = \begin{pmatrix} 2&2\\ 2&4 \end{pmatrix}$$

The skew-symmetric part is:

$$\frac{1}{2}(A-A^T) = \frac{1}{2} \begin{pmatrix} 0&2\\ -2&0 \end{pmatrix} = \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}$$

Therefore:

$$A= \begin{pmatrix} 2&2\\ 2&4 \end{pmatrix} + \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}$$

पहला matrix symmetric है और दूसरा matrix skew-symmetric है। दोनों को add करने पर original matrix $A$ वापस मिलती है।

Important Examination Points

• If $A$ is of order $m\times n$, then $A^T$ is of order $n\times m$.
• Transpose is denoted by $A^T$.
• $(A^T)^T=A$.
• $(A+B)^T=A^T+B^T$.
• $(A-B)^T=A^T-B^T$.
• $(kA)^T=kA^T$.
• $(AB)^T=B^TA^T$.
• A symmetric matrix satisfies $A^T=A$.
• A skew-symmetric matrix satisfies $A^T=-A$.
• Every skew-symmetric matrix has zero diagonal elements.
• Every square matrix can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix.

Multiple Choice Questions

1. If $A$ is of order $2\times3$, then the order of $A^T$ is:

   • (A) $2\times3$
   • (B) $3\times2$
   • (C) $2\times2$
   • (D) $3\times3$

2. A matrix $A$ is symmetric if:

   • (A) $A^T=-A$
   • (B) $A^T=A$
   • (C) $A^T=I$
   • (D) $A^T=O$

3. A skew-symmetric matrix satisfies:

   • (A) $A^T=A$
   • (B) $A^T=-A$
   • (C) $A^T=I$
   • (D) $A^T=O$

4. The diagonal elements of a skew-symmetric matrix are:

   • (A) $1$
   • (B) $-1$
   • (C) $0$
   • (D) Arbitrary

5. Which of the following is correct?

   • (A) $(AB)^T=A^TB^T$
   • (B) $(AB)^T=B^TA^T$
   • (C) $(AB)^T=AB$
   • (D) $(AB)^T=-AB$

Fill in the Blanks

1. If $A$ is $m\times n$, then $A^T$ is of order ______.
2. A symmetric matrix satisfies ______.
3. A skew-symmetric matrix satisfies ______.
4. For a skew-symmetric matrix, all diagonal elements are ______.
5. $(A+B)^T=$ ______.
6. $(AB)^T=$ ______.

True or False

1. The transpose of a square matrix always has the same order.
2. Every symmetric matrix is square.
3. Every skew-symmetric matrix has zero diagonal elements.
4. $(AB)^T=A^TB^T$.
5. $(A^T)^T=A$.
6. Every square matrix can be written as the sum of a symmetric matrix and a skew-symmetric matrix.

Practice Questions

1. Find the transpose of $$A=\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}.$$
2. Find $A^T$ for $$A=\begin{pmatrix}2&4\\5&7\end{pmatrix}.$$
3. Check whether $$A=\begin{pmatrix}2&3\\3&5\end{pmatrix}$$ is symmetric.
4. Check whether $$A=\begin{pmatrix}0&4\\-4&0\end{pmatrix}$$ is skew-symmetric.
5. Show that $(A+B)^T=A^T+B^T$ for suitable matrices.
6. Show that $(AB)^T=B^TA^T$ for suitable matrices.
7. Find the symmetric and skew-symmetric parts of $$A=\begin{pmatrix}3&5\\1&2\end{pmatrix}.$$
8. Determine whether $$A=\begin{pmatrix}1&2&3\\2&4&5\\3&5&6\end{pmatrix}$$ is symmetric.

Quick Revision

Concept	Important Result
Transpose	$A^T=[a_{ji}]$
Order	$A_{m\times n}\Rightarrow A^T_{n\times m}$
Transpose of Transpose	$(A^T)^T=A$
Transpose of Sum	$(A+B)^T=A^T+B^T$
Transpose of Difference	$(A-B)^T=A^T-B^T$
Transpose of Scalar Multiple	$(kA)^T=kA^T$
Transpose of Product	$(AB)^T=B^TA^T$
Symmetric Matrix	$A^T=A$
Skew-Symmetric Matrix	$A^T=-A$
Diagonal Elements of Skew-Symmetric Matrix	$a_{ii}=0$
Matrix Decomposition	$A=\frac{1}{2}(A+A^T)+\frac{1}{2}(A-A^T)$