Linear Algebra: Polynomial Interpolation

Today, in class, I answered some homework questions from Section 1.4, including the following problems:

  1. Show that an upper triangular matrix with non-zero diagonal entries is nonsingular. (We’ve not covered determinants yet so we used a row-equivalence to the Identity argument)
  2. Show that the inverse of an upper triangular matrix is upper triangular. (We used the fact that the same row operations that reduce a matrix to the identity will reduce the identity to the inverse of a matrix. The row operations that reduce an upper triangular matrix to the identity will necessarily change the identity into an upper triangular matrix.)
  3. Given the matrices, A and C, solve the matrix equation: XA + C = X.

The last homework example was over the Vandermonde matrix system, proving that the Vandermonde system is equivalent to polynomial interpolation. We also proved that if the values of the \mathbf{x}-vector (independent variables) are distinct, then the Vandermonde matrix is non-singular. As you might imagine, I got very excited and energetic about explaining this example. Polynomial interpolation was one of the first topics from Numerical Analysis that I fell “in love” with. It set me down the path to becoming a numerical analyst.

Originally, we had a text scheduled for the next class but because I spent a significant portion of class time on polynomial interpolation, I decided to postpone the exam until after I can answer some questions from the review and last section’s homework. We’ll do this next time and if there are no questions we’ll move on to the next chapter: a short one on determinants.

2 thoughts on “Linear Algebra: Polynomial Interpolation

  1. hi, i’m moena, mathematic student of Diponegoro university in semarang, west java Indonesia. i want to ask you, how to solve polynomial matrix diophantine equation, like A(s)X(s)+B(s)Y(S)=C(s), where X(s) and Y(s) is unknown. Thanks for your answer
    moena

  2. Hi.I am a Computer Science student.
    please insert here how u proof the first (#1) problem:Show that an upper triangular matrix with non-zero diagonal entries is nonsingular.
    thank u!

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