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Decompositions of a matrix by means of its dual matrices with applications☆

Research paper by Ik-PyoKima, Arnold R.Kräuterb

Indexed on: 10 Nov '17Published on: 01 Jan '18Published in: Linear Algebra and its Applications



Abstract

We introduce the notion of dual matrices of an infinite matrix A, which are defined by the dual sequences of the rows of A and naturally connected to the Pascal matrix P=[(ij)](i,j=0,1,2,…)<math class="math"><mi is="true">P</mi><mo is="true">=</mo><mo stretchy="false" is="true">[</mo><mrow is="true"><mo is="true">(</mo><mtable is="true"><mtr is="true"><mtd is="true"><mi is="true">i</mi></mtd></mtr><mtr is="true"><mtd is="true"><mi is="true">j</mi></mtd></mtr></mtable><mo is="true">)</mo></mrow><mo stretchy="false" is="true">]</mo><mtext is="true"></mtext><mo stretchy="false" is="true">(</mo><mi is="true">i</mi><mo is="true">,</mo><mi is="true">j</mi><mo is="true">=</mo><mn is="true">0</mn><mo is="true">,</mo><mn is="true">1</mn><mo is="true">,</mo><mn is="true">2</mn><mo is="true">,</mo><mo is="true">…</mo><mo stretchy="false" is="true">)</mo></math>. We present the Cholesky decomposition of the symmetric Pascal matrix by means of its dual matrix. Decompositions of a Vandermonde matrix are used to obtain variants of the Lagrange interpolation polynomial of degree ≤n that passes through the n+1<math class="math"><mi is="true">n</mi><mo is="true">+</mo><mn is="true">1</mn></math> points (i,qi)<math class="math"><mo stretchy="false" is="true">(</mo><mi is="true">i</mi><mo is="true">,</mo><msub is="true"><mrow is="true"><mi is="true">q</mi></mrow><mrow is="true"><mi is="true">i</mi></mrow></msub><mo stretchy="false" is="true">)</mo></math> for i=0,1,…,n<math class="math"><mi is="true">i</mi><mo is="true">=</mo><mn is="true">0</mn><mo is="true">,</mo><mn is="true">1</mn><mo is="true">,</mo><mo is="true">…</mo><mo is="true">,</mo><mi is="true">n</mi></math>.