Explain briefly what is wrong with the following statements: (i) An n × n matrix

Question

Explain briefly what is wrong with the following statements: (i) An n × n matrix is diagonalisable if and only if it has n distinct eigenvalues. (i) Every square matrix has an LU factorisation. Gii) Cholesky decompositions can be performed on every square matrix that is symmetric. (Giv) If a system of equations is not strictly dominant then neither the Gauss-Siedel nor the Jacobi method will converge to the solution. (v) Relaxation methods (in Linear Algebra) are methods used to improve the convergence of the Jacobi method.

Explanation / Answer

(i) An nxn matrix is diagonalisable if and only if it has n distinct eigenvalues.

There is nothing wrong in this statement.

(ii) Every square matrix has an LU factorization.

This statement is wrong.

Reason : Any square matrix admits LU factorization if the matrix is invertible.

(iii) Cholesky decompositions can be performed on every square matrix that is symmetric.

This statement is wrong.

Reason : Cholesky decompositions can be performed on every square matrix that is symmetric and positive definite.

(iv) If a system of equations is not strictly dominant then neither the Gauss-Seidel nor the Jacobi method will converge to the solution.

There is nothing wrong in this statement.

Reason : The Gauss-Jacobi iteration converges if and only if the system of equation is strictly diagonally dominant.

Again, the sufficient condition for the convergence of the Gauss-Seidel iteration method is that the system of equations must be strictly diagonally dominant.

(v) Relaxation method are methods used to improve the convergence of the Jacobi method.

There is nothing wrong in this statement.

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