Skew Symmetric Matrix

A square matrix A is called a skew symmetric matrix if the transpose of A is equal to the negative of A. In other words, if AT = -A, then matrix A is skew symmetric.

Explanation

To understand why a matrix satisfying AT = -A is called a skew symmetric matrix, let's break down the terms.

Square Matrix
A square matrix is a matrix that has the same number of rows and columns. In other words, it is a matrix where the number of rows is equal to the number of columns.

Transpose of a Matrix
The transpose of a matrix is obtained by interchanging its rows and columns. If A is a matrix, then the transpose of A is denoted by AT.

Negative of a Matrix
To find the negative of a matrix, we simply negate each element of the matrix. If A is a matrix, then the negative of A is denoted by -A.

Skew Symmetric Matrix
A square matrix A is said to be skew symmetric if its transpose is equal to the negative of the matrix itself. Mathematically, if AT = -A, then A is skew symmetric.

Example
Let's consider a 3x3 matrix A:

A = [a11 a12 a13
a21 a22 a23
a31 a32 a33]

The transpose of A is:

AT = [a11 a21 a31
a12 a22 a32
a13 a23 a33]

If A is skew symmetric, then AT = -A:

[a11 a21 a31
a12 a22 a32
a13 a23 a33] =
[-a11 -a21 -a31
-a12 -a22 -a32
-a13 -a23 -a33]

From this equation, we can see that the elements on the main diagonal of A are 0 because each element is equal to its negation. The elements above the main diagonal are negations of the corresponding elements below the main diagonal.

Therefore, a matrix A satisfying AT = -A is called a skew symmetric matrix.