Matrix Diagonalization

• is said to be diagonalizable if there exists an ordered basis for such that is a diagonal Matrix.

  is said to be diagonalizable if is similar to a diagonal matrix.

• (Friedberg 5.3) Let and a basis for . Then is diagonalizable if and only if is a diagonalizable matrix.

  It holds because linear transformations have corresponding matrices.

  It immediately follows that (Friedberg 5.3.1) A matrix is diagonalizable if and only if is diagonalizable.

• (Friedberg 5.4) is diagonalizable if and only if there exists a basis for and scalars (not necessarily distinct) such that such that

• Let be a linear operator on . A nonzero element is called an eigenvector of if there exists a scalar called the eigenvalue such that

  The set of eigenvalues is called the spectrum, denoted or for matrices. In this set, repetition is allowed.

• (Friedberg 5.7) A scalar is an eigenvalue of if and only if

  See Determinant

  • (Friedberg 5.7.1) Let . Then is an eigenvalue of if and only if
  • (Friedberg 5.7.2) Let be a linear operator on and a basis for . Then is an eigenvalue of if is an eigenvalue of

• (Friedberg e5.1.15) Let be a linear operator on and be an eigenvector of corresponding to eigenvalue . For any positive integer , is an eigenvector of corresponding to eigenvalue

• (Friedberg 5.10.1) Let be a linear operator on , where If has distinct eigenvalues then is diagonalizable

• A linear operator is diagonalizable if the following hold.

  • The corresponding characteristic polynomial of splits.

• (Friedberg e7.1.7e) is diagonalizable if and only if for each eigenvalue

• (Friedberg e5.2.11) If is invertible, then is diagonalizable if and only if is diagonalizable.

• (Friedberg e5.2.12) Let . Then is diagonalizable if and only if is diagonalizable.

Simultaneous Diagonalizability §

• Two linear operators on the same finite dimensional vector space are called simultaneously diagonalizable if there exists a basis for such that both and are diagonal matrices.

  Similarly, are simultaneously diagonalizable if there exists an invertible matrix such that both and are diagonal.

• (Friedberg e5.2.16) If and are simultaneously diagonalizable then

  (Friedberg e5.4.25) If and are diagonalizable linear operators such that , then and are simultaneously diagonalizable. *

• (Friedberg e5.2.17) and are simultaneously diagonalizable for any .

• (Friedberg e6.4.14) Let be a finite dimensional real inner product space and are self-adjoint operators on such that . There exists an orthonormal basis for consisting of vectors that are eigenvectors of both and .

• (Friedberg e6.6.10) If are normal operators such that , there exists an orthonormal basis for consisting of vectors that are eigenvectors of both and .

Topics §

• Characteristic Polynomial
• Eigenspaces and Eigenbases
• Spectral Theorem
• Jordan Canonical Form
• Rational Canonical Form

Links §

• Friedberg, Insel and Spence - Ch. 5