Eigenvalues and Eigenvectors

This chapter systematically studies the eigenvalues and eigenvectors of matrices, similarity, diagonalization, and properties of real symmetric matrices.

Definition of Eigenvalues and Eigenvectors

• $A\vec{x} = \lambda \vec{x}$
• $|A - \lambda I| = 0$ is the characteristic equation

Properties and Computation

• An $n$-order matrix has $n$ eigenvalues (counting multiplicities)
• Eigenvectors corresponding to different eigenvalues are linearly independent
• Eigenvectors corresponding to different eigenvalues are linearly independent

Similarity and Diagonalization

• Similarity: $B = P^{-1}AP$
• Diagonalizability condition: there are $n$ linearly independent eigenvectors
• Real symmetric matrices can be orthogonally diagonalized

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Exercises

1. Find the eigenvalues and eigenvectors of $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$.
2. Determine whether $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ is diagonalizable.
3. If $A$ has eigenvalues $1,2,3$, write $|A|$ and $\operatorname{tr}A$.
4. Determine whether $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is a real symmetric matrix, and whether all its eigenvalues are real.
5. If $A$ is diagonalizable, write its diagonalization form.

Reference Answers

1. Eigenvalues and eigenvectors

$|A - \lambda I| = (2-\lambda)^2 - 1 = \lambda^2 - 4\lambda + 3 = 0$, $\lambda=1,3$

For $\lambda=1$, $\vec{x} = (1,-1)$; for $\lambda=3$, $\vec{x} = (1,1)$

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2. Diagonalizability

There is only one eigenvector, so it is not diagonalizable.

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3. $|A|$ and $\operatorname{tr}A$

$|A| = 1\times2\times3 = 6$, $\operatorname{tr}A = 1+2+3=6$

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4. Real symmetry and eigenvalues

$A$ is a real symmetric matrix, eigenvalues are $1,-1$, both real.

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5. Diagonalization form

$A = PDP^{-1}$, $D$ is a diagonal matrix, $P$ is the matrix of eigenvectors.

上一章节 Systems of Linear EquationsUnderstand the structure of solutions for homogeneous and nonhomogeneous systems, master Cramer's rule, fundamental solution sets, general solutions, and elementary row transformations. 下一章节 Quadratic FormsUnderstand the definition, matrix representation, canonical form, positive definiteness, and congruence transformation of quadratic forms.