Hurkyl said:It seems to me that the obvious first thing to do would be to look at the transpose of (M^-1)*A*M.
A symmetric matrix is a square matrix where the elements above and below the main diagonal are reflections of each other. In other words, if we flip the matrix along its main diagonal, the elements will remain in the same positions.
An orthogonal matrix is a square matrix where the columns and rows are orthogonal to each other. This means that the dot product of any two columns or rows is equal to 0.
If an orthogonal matrix, say Q, is multiplied by a symmetric matrix, say A, the resulting matrix will also be symmetric. This is because the dot product of any two columns or rows in QA is equal to the dot product of the corresponding columns or rows in A. Since A is symmetric, these dot products will be equal, making QA also symmetric.
No, an orthogonal matrix can only prove that a matrix is symmetric if both matrices are square. This is because in order for matrix multiplication to be valid, the number of columns in the first matrix must match the number of rows in the second matrix.
Proving a symmetric matrix with an orthogonal matrix is useful because it allows us to prove the symmetry of a matrix without having to manually check each element. It also helps us understand the relationship between symmetry and orthogonal matrices, which has applications in various fields such as physics, engineering, and computer science.