In: Advanced Math
Explain how the rank of a matrix and existence and uniqueness of solutions of “Systems of Linear Equations” are related
.
Explain how Eigenfunctions”, “Eigenvalues” and “Orthogonality” terms and concepts are defined in Matrix algebra.
# Let A be any matrix of order nm.
The rank of a matrix is the number of non-zero rows in row echelon form of matrix.
By writing system of linear equations in matrix form, let [ A | D ] denotes augmented matrix.
The solution exists & it is unique if & only if Rank (A ) = Rank ( A| D ).
# Let A be any square matrix of order n.
Eigenvalue :The eigenvalue is defined
to be root of characteristics polynomial of A .
i.e. To find eigenvalue we will solve
determinant ( A -
I ) = 0 , then
we will get characteristics equation & its roots are called as
eigenvalues of A.
Eigenfunctions :To find eigenfunctions
corresponding to eigenvalue , we have to
solve the system (A -
I ) X = 0.
where I is identity matrix of order n. The solution of this
system are eigenfunctions corresponding to eigenvalue .
Orthogonality : A matrix "A" is said to be orthogonal if A.At = At.A = I . (At denotes transpose of A)