The section 8.4 of the uploaded document discusses Eigenbasis and Diagonalization in the context of matrix eigenvalue problems. Here's a breakdown of the content:

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Eigenbasis:

1. Definition:

   • An eigenbasis refers to a basis formed by eigenvectors of a matrix.
   • A matrix $A$ of order $n$ can have $n$ distinct eigenvectors if it has $n$ distinct eigenvalues.
   • These eigenvectors are linearly independent (L.I.) and span the space $\mathbb{R}^n$ or $\mathbb{C}^n$.

2. Properties:

   • Symmetric matrices have an orthonormal basis of eigenvectors.
   • Hermitian, skew-Hermitian, or unitary matrices possess a basis of eigenvectors that form a unitary system.

3. Condition for Eigenbasis:

   • A basis of eigenvectors exists if the algebraic multiplicity (number of times an eigenvalue appears in the characteristic equation) equals the geometric multiplicity (number of linearly independent eigenvectors corresponding to the eigenvalue).

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Diagonalization:

1. Similarity Transformation:

   • A matrix $A$ is diagonalizable if there exists a nonsingular matrix $P$ such that: $P^{-1}AP = D$ where $D$ is a diagonal matrix whose diagonal elements are the eigenvalues of $A$.

2. Procedure:

   • Compute eigenvalues of $A$.
   • Find corresponding eigenvectors.
   • Form the modal matrix $X$, where each column is an eigenvector of $A$.
   • Compute $X^{-1}$, the inverse of the modal matrix.
   • Use $D = X^{-1}AX$ to diagonalize.

3. Diagonal Matrix:

   • The diagonal elements of $D$ are the eigenvalues of $A$.

4. Key Notes:

   • Only matrices with a full set of linearly independent eigenvectors can be diagonalized.
   • Powers of $A$ can be computed using $A^n = XD^nX^{-1}$, where $D^n$ is the diagonal matrix with eigenvalues raised to the power $n$.

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Example:

The section provides a worked example to diagonalize a matrix. It involves:

• Finding eigenvalues.
• Determining eigenvectors.
• Constructing the modal matrix $X$ and its inverse $X^{-1}$.
• Verifying diagonalization using $A = XD X^{-1}$.

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Let me know if you'd like detailed solutions to specific examples or additional clarifications.

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Related Questions:

1. What are the steps to find the eigenvalues and eigenvectors of a matrix?
2. Why must the algebraic multiplicity equal the geometric multiplicity for diagonalization?
3. How do symmetric and Hermitian matrices guarantee orthonormal eigenbases?
4. What is the role of the modal matrix in diagonalization?
5. How can diagonalization simplify computing powers of a matrix?

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Tip:

When diagonalizing, always verify that the eigenvectors are linearly independent, as this determines if the matrix is truly diagonalizable.