Calculator Input
Enter a square matrix, choose tolerance settings, and evaluate whether the matrix equals the negative of its transpose.
Example Data Table
| Example | Matrix | Expected Result | Reason |
|---|---|---|---|
| Example 1 | [ [0, 2, -1], [-2, 0, 4], [1, -4, 0] ] | Pass | Transpose equals the negative matrix, and the diagonal is zero. |
| Example 2 | [ [0, 3], [-3, 0] ] | Pass | Both off-diagonal entries are exact negatives of each other. |
| Example 3 | [ [1, 2], [-2, 0] ] | Fail | The diagonal entry is not zero. |
| Example 4 | [ [0, 2], [2, 0] ] | Fail | Off-diagonal entries do not have opposite signs. |
Formula Used
Main test: A matrix A is skew symmetric when AT = -A.
Equivalent residual test: A + AT = 0.
Diagonal condition: Every diagonal entry must satisfy aii = 0.
Pair condition: For each off-diagonal pair, aij = -aji.
Tolerance rule: The calculator accepts values when |aij + aji| ≤ tolerance and |aii| ≤ tolerance.
How to Use This Calculator
- Select the matrix order from 2×2 to 6×6.
- Enter a tolerance if you want decimal rounding flexibility.
- Choose the number of display decimals for tables and summaries.
- Type values into the matrix grid or paste them into the import box.
- Use the example buttons if you want a ready-made test matrix.
- Press Check Matrix to generate the result section above the form.
- Review the decision, residual matrix, mismatch list, and Plotly heatmap.
- Download a CSV or PDF report when you need to save the analysis.
FAQs
1) What makes a matrix skew symmetric?
A matrix is skew symmetric when its transpose equals its negative. Every diagonal value must be zero, and each off-diagonal pair must have equal magnitude with opposite sign.
2) Why must the diagonal entries be zero?
For any diagonal entry, the skew symmetry rule gives aii = -aii. That only happens when aii equals zero, unless you allow a small tolerance for rounding.
3) Does the matrix have to be square?
Yes. Transpose comparisons require the same number of rows and columns. A non-square matrix cannot satisfy the condition AT = -A for the entire matrix.
4) What does the tolerance setting do?
Tolerance allows tiny decimal errors during comparison. This is useful when matrix values come from measurements, floating-point calculations, or imported data with small rounding differences.
5) What does the residual matrix show?
The residual matrix is A + AT. A perfect skew symmetric matrix produces zeros everywhere. Nonzero cells highlight exactly where the skew symmetry condition fails.
6) Can a zero matrix be skew symmetric?
Yes. The zero matrix satisfies AT = A and also AT = -A because every entry is zero. It is both symmetric and skew symmetric.
7) How is skew symmetric different from symmetric?
A symmetric matrix satisfies AT = A. A skew symmetric matrix satisfies AT = -A. Symmetric matrices mirror values, while skew symmetric matrices mirror opposite signs.
8) Why does this calculator show pair matches and norms?
Pair matches show how many off-diagonal comparisons passed. The Frobenius residual norm summarizes total error magnitude, helping you judge how close an almost-skew matrix is to the target structure.