mlarson9000 said:(A+A^T)is symmetric, (A-A^T)is skew symmetric, but adding them together produces 2A, not A. I'm not sure what to do with this information.
Dick said:That's not such a big problem. Divide each one by two.
The unique decomposition of a square matrix is a way to express a square matrix as a product of two matrices, one of which is a lower triangular matrix and the other is an upper triangular matrix. This decomposition is also known as LU decomposition.
Proving the unique decomposition of a square matrix is important because it allows us to solve systems of linear equations efficiently. This decomposition reduces the complexity of solving a system of equations from O(n^3) to O(n^2), making it a valuable tool in many scientific and engineering applications.
The unique decomposition of a square matrix is calculated using Gaussian elimination, a method of reducing a matrix to echelon form. The lower triangular matrix is found by performing elementary row operations on the original matrix until it is in echelon form, and the upper triangular matrix is found by performing the same operations on the transpose of the original matrix.
No, not every square matrix can be uniquely decomposed. A square matrix can only be uniquely decomposed if it is non-singular, meaning it has a non-zero determinant. If the matrix is singular, it cannot be decomposed and other methods must be used to solve the system of equations.
Aside from reducing the complexity of solving systems of linear equations, the unique decomposition of a square matrix also helps to identify if a matrix is invertible or not. It is also useful in numerical analysis and can improve the efficiency of algorithms such as the Cholesky decomposition and the QR factorization.