What properties must general orthogonal matrices possess?
Conditions
- The matrix is classified as an orthogonal matrix.
Reasoning, step by step
- Check the length of each column vector; it must be one (unit-length).
- Check the dot product between any two distinct column vectors; it must be zero (mutually perpendicular).
- Conclude that satisfying both conditions makes the columns orthonormal.
Example
General orthogonal matrices require unit-length as well as mutually perpendicular columns.
Common misconceptions
- Thinking that mutually perpendicular columns are sufficient without requiring unit length.
- Believing that unit-length columns are sufficient without requiring mutual perpendicularity.
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