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Eigenvalues and Eigenvectors

Part 4: The Trace Operation

  1. Display each of the matrices A, B, and C, together with its trace. Next enter the diagonal matrix F, and display its trace as well. By comparing each matrix to its trace, formulate a rule on how the trace may be calculated in terms of the entries of the matrix.
  2. Use a random square matrix G to check your rule. Recompute the random matrix, and test your rule several times.
  3. Now look at the eigenvalues of the matrices A, B, C, and F, and compare them to the corresponding trace of each matrix. Formulate a rule relating the eigenvalues of a matrix to its trace.
  4. Check your rule with random symmetric matrices of varying sizes.
  5. Check your rule with random matrices of varying sizes, not necessarily symmetric.
  6. Explain why your rule must be true for for the special case of diagonal matrices.

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