No, not all symmetric matrices are orthogonal.
Understanding the Definitions
- Symmetric Matrix: A square matrix where the elements across the main diagonal are equal. For example:
[ 1 2 ] [ 2 3 ] - Orthogonal Matrix: A square matrix where the transpose of the matrix is also its inverse. In other words, the dot product of any two distinct columns (or rows) is zero. For example:
[ cos(θ) -sin(θ) ] [ sin(θ) cos(θ) ]
Why Not All Symmetric Matrices are Orthogonal
A symmetric matrix can have eigenvalues that are not equal to 1 or -1. For a matrix to be orthogonal, all its eigenvalues must be either 1 or -1.
Example
Consider the symmetric matrix:
[ 2 0 ]
[ 0 1 ]This matrix is symmetric but not orthogonal because its eigenvalues are 2 and 1, which are not equal to 1 or -1.
Key Points
- A matrix can be symmetric without being orthogonal.
- Orthogonality implies specific properties of the matrix's eigenvalues.
- Symmetric matrices have special properties in linear algebra, but orthogonality is not one of them.
