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Linear Algebra Chapter 6

Linear Algebra Chapter 6. Linear Algebra with Applications -Gareth Williams Br. Joel Baumeyer, F.S.C. Definition: Eigenvalue & Eigenvector. The Characteristic Polynomial of A nxn : |A -  I | The Characteristic Equation of A nxn : |A -  I | = 0. Theorem 6.1.

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Linear Algebra Chapter 6

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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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