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Understanding Eigenvalues and Eigenvectors in Linear Algebra

Explore the concepts of eigenvalues and eigenvectors in linear algebra, including similarity transformations and diagonalizable matrices. Learn how to demonstrate powers of diagonalizable matrices and understand properties of symmetric matrices.

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Understanding Eigenvalues and Eigenvectors in Linear Algebra

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  1. Linear Algebra Chapter 6 Linear Algebra with Applications -Gareth Williams Br. Joel Baumeyer, F.S.C.

  2. Definition: Eigenvalue & Eigenvector • The Characteristic Polynomial of Anxn: |A - I| • The Characteristic Equation of Anxn: |A - I| = 0

  3. Theorem 6.1

  4. Similar Matrices • Definition: Let A and B be square matrices of the same size. B is said to be similar to A if there exists an invertible matrix C such that B = C-1AC. The transformation of the matrix A into the matrix B in this manner is called a similarity transformation.

  5. Theorem 6.3 Theorem 6.2 is in the optional section 6.2 and not covered.

  6. Diagonalizable Matrix • A square matrix A is said to be diagonalizable if there exists a matrix C such that D = C-1AC is a diagonal matrix.

  7. Theorem 6.4 • Let A be an nn matrix. • (a) If A has n linearly independent eigenvectors, it is diagonalizable. The matrix C whose columns consist of n linearly independent eigenvectors can be used in a similarity transformation C-1AC to give a diagonal matrix d. The diagonal elements of D will be the eigenvalues of A. • (b) If A is diagonalizable, then it has n linearly independent eigenvectors.

  8. Powers of Diagonalizable Matrices • If A is similar to a diagonal mamatrix D under the transformation C-1AC, then it can be shown that Ak = CDkC-1. • Demonstration: Dk = (C-1AC)k = (C-1AC) … (C-1AC) = (C-1AkC) and reversing gives: Ak = CDkC-1

  9. Theorem 6.5 • Let a be an nxn symmetrix matrix. • a) All the eigenvalues of A are real numbers. • b) the dimension of an eigenspace of A is the multiplicity of the eigenvalue as a root of the characteristic equation. • c) The eigenspaces of A are orthogonal. • d)A has n linearly independent eigenvectors.

  10. Orthogonally Diagonalizable • Definition: A square matrix A is said to be orthogonally diagonalizable if there exists an ortholgonal matrix C such that D = CtAC is a diagonal matrix.

  11. Theorem 6.6 • Let a be a square matrix. A is orthog-onally diagonalizable if and only if it is a symmetric matrix.

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