# Linear algebra ( symmetric matrix)

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• Vijay Raghavan
In summary, for any matrix A, A*At and (A+At)/2 are both symmetric matrices, and the factor of 1/2 is not necessary when A is a sum of a symmetric and skew-symmetric matrix.
Vijay Raghavan
I am currently brushing on my linear algebra skills when i read this
For any Matrix A
1)A*At is symmetric , where At is A transpose ( sorry I tried using the super script option given in the editor and i couldn't figure it out )
2)(A + At)/2 is symmetric
Now my question is , why should it be divided by 2? doesn't just A + At alone give a symmetric matrix

Vijay Raghavan said:
I am currently brushing on my linear algebra skills when i read this
For any Matrix A
1)A*At is symmetric , where At is A transpose ( sorry I tried using the super script option given in the editor and i couldn't figure it out )
2)(A + At)/2 is symmetric
Now my question is , why should it be divided by 2? doesn't just A + At alone give a symmetric matrix
It goes like this ##\text{ $A^t$ }## or ##\text{ ## A^t ## }##.

The relevant formulas are ##(A \cdot B)^t = B^t \cdot A^t \, , \, (A+B)^t = A^t + B^t ## and ##(A^t)^t=A##.
You are correct, the factor ##\frac{1}{2}## isn't necessary here. It usually is taken when ##A## is written as ##A = \frac{1}{2}(A+A^t) + \frac{1}{2}(A-A^t)##, i.e. as a sum of a symmetric matrix ##B=B^t=\frac{1}{2}(A+A^t)## and a skew-symmetric matrix ##C=-C^t=\frac{1}{2}(A-A^t)##. Here it is needed to get back ##A##, instead of ##2A##.

Vijay Raghavan
fresh_42 said:
It goes like this ##\text{ $A^t$ }## or ##\text{ ## A^t ## }##.

The relevant formulas are ##(A \cdot B)^t = B^t \cdot A^t \, , \, (A+B)^t = A^t + B^t ## and ##(A^t)^t=A##.
You are correct, the factor ##\frac{1}{2}## isn't necessary here. It usually is taken when ##A## is written as ##A = \frac{1}{2}(A+A^t) + \frac{1}{2}(A-A^t)##, i.e. as a sum of a symmetric matrix ##B=B^t=\frac{1}{2}(A+A^t)## and a skew-symmetric matrix ##C=-C^t=\frac{1}{2}(A-A^t)##. Here it is needed to get back ##A##, instead of ##2A##.
Thank you for the clarification.

## 1. What is a symmetric matrix?

A symmetric matrix is a square matrix that is equal to its own transpose. In other words, the elements along the main diagonal (from top left to bottom right) remain unchanged, while the elements above and below the main diagonal are reflections of each other.

## 2. What are the properties of a symmetric matrix?

Some key properties of a symmetric matrix include:

• It is always square, with the same number of rows and columns.
• It is equal to its own transpose, meaning that A = AT.
• The eigenvalues of a symmetric matrix are always real numbers.
• It is always diagonalizable, meaning it can be expressed as a diagonal matrix using an orthogonal matrix.

## 3. How is a symmetric matrix used in linear algebra?

Symmetric matrices are used in many applications of linear algebra, such as in optimization problems, diagonalization of matrices, and solving systems of linear equations. They also have important applications in geometry, physics, and engineering.

## 4. Can a non-square matrix be symmetric?

No, a non-square matrix cannot be symmetric. Since a symmetric matrix must be equal to its own transpose, it must have the same number of rows and columns, making it a square matrix.

## 5. How can I determine if a matrix is symmetric?

To determine if a matrix is symmetric, you can check if it is equal to its own transpose. This means comparing each element in the matrix with its corresponding element in the transpose. If they are all equal, then the matrix is symmetric. Another way is to check if the matrix is equal to its reflection across the main diagonal, since this is a key property of symmetric matrices.

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