To determine if a matrix is skew symmetric, you check if its transpose is equal to the negative of the original matrix. This fundamental property, Aᵀ = -A, defines a skew-symmetric matrix.
Here's the step-by-step process to check if a given matrix is skew-symmetric:
Steps to Check for Skew-Symmetry
According to the definition and common practice:
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Step 1: Find the transpose of the originally given matrix.
The transpose of a matrix A, denoted by Aᵀ or A', is obtained by interchanging its rows and columns. If A is an m x n matrix, then Aᵀ is an n x m matrix where the element in the i-th row and j-th column of Aᵀ is the element in the j-th row and i-th column of A. For a square matrix (which skew-symmetric matrices must be), if A = [aᵢⱼ], then Aᵀ = [aⱼᵢ]. -
Step 2: Find the negative of the original matrix.
The negative of a matrix A, denoted by -A, is found by multiplying every element in the matrix by -1. If A = [aᵢⱼ], then -A = [-aᵢⱼ]. -
Step 3: Compare the transpose and the negative matrix.
If the transpose of the matrix obtained in Step 1 (Aᵀ) is equal to the negative of the original matrix obtained in Step 2 (-A), then the matrix is said to be skew-symmetric. That is, A is skew-symmetric if and only if Aᵀ = -A.
Key Condition: A matrix A is skew-symmetric if Aᵀ = -A.
This also implies that for every element aᵢⱼ in the matrix A, the corresponding element aⱼᵢ in the transpose must be equal to -aᵢⱼ. So, aⱼᵢ = -aᵢⱼ for all i and j.
Example: Checking a Matrix for Skew-Symmetry
Let's consider a 3x3 matrix A:
A = $\begin{pmatrix} 0 & 2 & -1 \ -2 & 0 & 3 \ 1 & -3 & 0 \end{pmatrix}$
Now, let's follow the steps:
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Find the transpose (Aᵀ):
We swap rows and columns.
Aᵀ = $\begin{pmatrix} 0 & -2 & 1 \ 2 & 0 & -3 \ -1 & 3 & 0 \end{pmatrix}$ -
Find the negative of the original matrix (-A):
Multiply each element of A by -1.
-A = $\begin{pmatrix} -(0) & -(2) & -(-1) \ -(-2) & -(0) & -(3) \ -(1) & -(-3) & -(0) \end{pmatrix}$ = $\begin{pmatrix} 0 & -2 & 1 \ 2 & 0 & -3 \ -1 & 3 & 0 \end{pmatrix}$ -
Compare Aᵀ and -A:
Aᵀ = $\begin{pmatrix} 0 & -2 & 1 \ 2 & 0 & -3 \ -1 & 3 & 0 \end{pmatrix}$
-A = $\begin{pmatrix} 0 & -2 & 1 \ 2 & 0 & -3 \ -1 & 3 & 0 \end{pmatrix}$Since Aᵀ = -A, the matrix A is indeed skew-symmetric.
You can see this clearly in a comparison table:
| Element Position | Original Matrix (aᵢⱼ) | Transpose Matrix (aⱼᵢ) | Negative Matrix (-aᵢⱼ) | Check (aⱼᵢ = -aᵢⱼ) |
|---|---|---|---|---|
| (1,1) | 0 | 0 | 0 | 0 = 0 (True) |
| (1,2) | 2 | -2 | -2 | -2 = -2 (True) |
| (1,3) | -1 | 1 | 1 | 1 = 1 (True) |
| (2,1) | -2 | 2 | 2 | 2 = 2 (True) |
| (2,2) | 0 | 0 | 0 | 0 = 0 (True) |
| (2,3) | 3 | -3 | -3 | -3 = -3 (True) |
| (3,1) | 1 | -1 | -1 | -1 = -1 (True) |
| (3,2) | -3 | 3 | 3 | 3 = 3 (True) |
| (3,3) | 0 | 0 | 0 | 0 = 0 (True) |
All elements satisfy the condition aⱼᵢ = -aᵢⱼ, confirming that the matrix is skew-symmetric.
Properties of Skew-Symmetric Matrices
Understanding these properties can help in identifying or working with skew-symmetric matrices:
- Diagonal Elements: The diagonal elements of a skew-symmetric matrix must always be zero. This is because for a diagonal element aᵢᵢ, the condition aᵢᵢ = -aᵢᵢ implies 2aᵢᵢ = 0, which means aᵢᵢ = 0.
- Square Matrix: A skew-symmetric matrix must be a square matrix (number of rows equals number of columns).
- Sum/Difference: The sum of a matrix A and its transpose Aᵀ is always a symmetric matrix (A + Aᵀ is symmetric). The difference between a matrix A and its transpose Aᵀ is always a skew-symmetric matrix (A - Aᵀ is skew-symmetric). Any square matrix can be uniquely expressed as the sum of a symmetric and a skew-symmetric matrix.
In summary, "solving skew symmetric" in this context means verifying the core relationship Aᵀ = -A through the steps of finding the transpose and the negative of the matrix and comparing them.
