Question

In: Advanced Math

Give a proof, base the proof on the Determinant of a Vandermonde matrix that the INTERPOLATING...

Give a proof, base the proof on the Determinant of a Vandermonde matrix that the INTERPOLATING POLYNOMIAL exist and its unique.

Expert Solution

Suppose that the interpolation matrix exists and is in the form :

Now, the statement that interpolates the data points mean that:

Substituting this in the first equation, we get:

We need to solve for the s to construct . Since such a construction is possible, by reversing the argument, we see that the interpolating polynomial exists. Also, the matrix on the left is called a Vandermonde matrix.

To prove it's uniqueness, we write the above matrix equation as:

We have that is non-singular, because:

because the points are distinct, the determinant can't be zero as is never zero, therefore is nonsingular and the system has a unique solution .

Colby Messinger answered 2 years ago

Find the adjoint of matrix A, the determinant of matrix A, and the determinant of the...

Find the adjoint of matrix A, the determinant of matrix A, and the determinant of the adjoint A. A= 1 1 0 2 2 1 1 0 0 2 1 1 1 0 2 1

Is there a relation between the determinant of the matrix associated to a linear mapping and...

Is there a relation between the determinant of the matrix associated to a linear mapping and the bijection of the mapping?

In economics, What is the role or function of the Determinant of a Vandermonde Matrix?

Suppose B is a (2x2) matrix such that B2 = I . Showthat the determinant...

Suppose B is a (2x2) matrix such that B2 = I . Show that the determinant I B I is either +1 or -1. [Hint: A2stands for the matrix product AA. I is the (2x2) identity matrix. A fact: For any two matrices E and F, I EF I = I E I I F I ]

Hessian Matrix Determinant and Convex/Concave I have a function which has a Hessian Matrix of: -2...

Hessian Matrix Determinant and Convex/Concave I have a function which has a Hessian Matrix of: -2 0 0 0 -16 10 0 10 -4 Can anyone explain why this is neither a positive nor a negative definite? And could you please explain 2*2 and 3*3 Hessian's rule of determining positive/negative definite? Thank you.

Matlab You will write a function to calculate the determinant of a matrix. It should work...

Matlab You will write a function to calculate the determinant of a matrix. It should work for any size matrix. Remember that the determinant can be calculated by multiplying the diagonal elements of an upper right triangular matrix. Your function will take a matrix passed to it and put it in upper right triangular form. You will work down the diagonal beginning at row 1 column 1, then row 2 column 2, etc. Note that the row and column numbers...

The determinant of a matrix is the product of its eigenvalues. Can you prove this when...

The determinant of a matrix is the product of its eigenvalues. Can you prove this when A is diagonalizable? How about if A is 2 x 2, and may or may not be diagonalizable? (Hint: What's the constant term in the characteristic polynomial>

Make a code in matlab to know the determinant of a matrix n x n, using...

Make a code in matlab to know the determinant of a matrix n x n, using the sarrus rule.

Compute the determinant of A, where A= a 4x4 matrix [1 -3 0 0; 2 1...

Compute the determinant of A, where A= a 4x4 matrix [1 -3 0 0; 2 1 0 0; 0 0 1 2; 0 0 2 1] a 4x4 matrix [2 5 4 2; 0 0 0 2; 0 -3 0 -4; 1 0 -1 1] and a 4x4 matrix [1 -3 0 0; 2 1 0 0; 0 0 1 2; 0 0 2 1]^-1. a) det(A)= -36 b) det(A)= 5 c) det(A)= 0 d) det(A)= -13 e)det(A)= 36

How to proof: Matrix A have a size of m×n, and the rank is r. How...

How to proof: Matrix A have a size of m×n, and the rank is r. How can we rigorous proof that the dimension of column space are always equal to the dimensional of row space? (I can use many examples to show this work, but how to proof rigorously?)