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Let K, M be n ×n matrices, with M>0 positive definite. A...

Let $K, M$ be $n \times n$ matrices, with $M>0$ positive definite. A nonzero vector $\mathbf{v} \neq \mathbf{0}$ is called a generalized eigenvector of matrix pair $K, M$ if it satisfies the generalized eigenvalue equation $$ K \mathbf{v}=\lambda M \mathbf{v}, \quad \mathbf{v} \neq \mathbf{0}, $$ where the scalar $\lambda$ is the corresponding generalized eigenvalue. Note that ordinary eigenvalue/eigenvectors of $K$ are when $M=1$.(a) Prove that $\lambda$ is a generalized eigenvalue if and only if it satisfies the generalized characteristic equation $\operatorname{det}(K-\lambda M)=0$. (b) Prove that $\lambda$ is a generalized eigenvalue of the matrix pair $K, M$ if and only if it is an ordinary eigenvalue of the matrix $M^{-1} K$. How are the eigenvectors related? (c) Now suppose $K$ is a symmetric matrix. Prove that its generalized eigenvalues are all real. Hint: First explain why this does not follow from part (a). Instead mimic the proof of part (a) of Theorem 8.32 , using the weighted Hermitian inner product $\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^T M \mathbf{w}$ in place of the dot product. (d) Show that if $K>0$, then its generalized eigenvalues are all positive: $\lambda>0$. (e) Prove that the eigenvectors corresponding to different generalized eigenvalues are orthogonal under the weighted inner product $\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^T M \mathbf{w}$. (f) Show Cuibuag Thascoag com that, if the matrix pair $K, M$ has $n$ distinct generalized eigenvalues, then the eigenvectors form an orthogonal basis for $\mathbb{R}^n$. Remark. One can, by mimicking the proof of part $(c)$ of Theorem 8.32 , show that this holds even when there are repeated generalized eigenvalues.

Let $K, M$ be $n \times n$ matrices, with $M>0$ positive definite. A nonzero vector $\mathbf{v} \neq \mathbf{0}$ is called a generalized eigenvector of matrix pair $K, M$ if it satisfies the generalized eigenvalue equation
$$
K \mathbf{v}=\lambda M \mathbf{v}, \quad \mathbf{v} \neq \mathbf{0},
$$
where the scalar $\lambda$ is the corresponding generalized eigenvalue. Note that ordinary eigenvalue/eigenvectors of $K$ are when $M=1$.(a) Prove that $\lambda$ is a generalized eigenvalue if and only if it satisfies the generalized characteristic equation $\operatorname{det}(K-\lambda M)=0$.
(b) Prove that $\lambda$ is a generalized eigenvalue of the matrix pair $K, M$ if and only if it is an ordinary eigenvalue of the matrix $M^{-1} K$. How are the eigenvectors related? (c) Now suppose $K$ is a symmetric matrix. Prove that its generalized eigenvalues are all real.
Hint: First explain why this does not follow from part (a). Instead mimic the proof of part (a) of Theorem 8.32 , using the weighted Hermitian inner product $\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^T M \mathbf{w}$ in place of the dot product. (d) Show that if $K>0$, then its generalized eigenvalues are all positive: $\lambda>0$. (e) Prove that the eigenvectors corresponding to different generalized eigenvalues are orthogonal under the weighted inner product $\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^T M \mathbf{w}$. (f) Show
Cuibuag Thascoag com
that, if the matrix pair $K, M$ has $n$ distinct generalized eigenvalues, then the eigenvectors form an orthogonal basis for $\mathbb{R}^n$. Remark. One can, by mimicking the proof of part $(c)$ of Theorem 8.32 , show that this holds even when there are repeated generalized eigenvalues.

Key Concepts

Weighted Inner Product

The weighted inner product ?v, w? = v?Mw is a key tool in studying generalized eigenvalue problems. It adapts the concept of orthogonality and norms to reflect the weighting introduced by the matrix M, thereby preserving the geometric and spectral properties analogous to those in the standard theory.

Orthogonality of Generalized Eigenvectors

Under the weighted inner product, eigenvectors corresponding to distinct generalized eigenvalues of the matrix pair (K, M) are orthogonal. This orthogonality is crucial for decomposing the space and for applications such as diagonalization, similar to the spectral theorem for symmetric matrices in the standard setting.

Symmetry and Real Eigenvalues

In the context of a symmetric matrix K, the use of a weighted inner product, defined by M, shows that the generalized eigenvalues are real. This is significant because symmetry in the standard eigenvalue problem guarantees real eigenvalues, and a similar result holds when extending to the generalized problem under the appropriate inner product.

Positive Definiteness

Positive definiteness of M (and K, in certain contexts) ensures that M is invertible and that the generalized eigenvalue problem behaves well. When K is also positive definite, it guarantees that all generalized eigenvalues are positive, which has important implications in stability analysis and physical interpretations of the system.

Generalized Characteristic Equation

The generalized characteristic equation, det(K - ?M) = 0, is the algebraic condition under which the generalized eigenvalue problem has nontrivial solutions. It parallels the standard eigenvalue condition by providing a necessary and sufficient criterion for the existence of eigenvalues corresponding to the matrix pair (K, M).

Generalized Eigenvalue Problem

This concept involves finding nonzero vectors v and scalars ? such that Kv = ?Mv, where M is a given positive definite matrix. This extends the standard eigenvalue problem by considering a pair of matrices and is essential in various applications where systems of equations are weighted or constrained by an additional matrix.

Matrix Transformation via Inverse

When M is positive definite, it is invertible, allowing the generalized eigenvalue problem Kv = ?Mv to be rewritten as M?¹K v = ?v. This transformation links the generalized problem to an ordinary eigenvalue problem for the matrix M?¹K, making many properties of the standard eigenvalue problem applicable in this broader context.

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