(a) Let A be an n ×n symmetric matrix, and let v be an eigenvector. Prove that its orthogonal co

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(a) Let A be an n ×n symmetric matrix, and let v be an...

(a) Let $A$ be an $n \times n$ symmetric matrix, and let $\mathrm{v}$ be an eigenvector. Prove that its orthogonal complement under the dot product, namely, $V^{\perp}=\left\{\mathbf{w} \in \mathbb{R}^n \mid \mathbf{v}_1 \cdot \mathbf{w}=0\right\}$, is an invariant subspace. (b) More generally, prove that if $W \subset \mathbb{R}^n$ is an invariant subspace, then its orthogonal complement $W^{\perp}$, is also invariant.

Key Concepts

Invariant Subspace

An invariant subspace of a linear transformation is a subspace that is mapped onto itself by that transformation. This means that if a vector belongs to an invariant subspace, its image under the transformation also lies in the same subspace. Recognizing and working with invariant subspaces is fundamental in the study of linear operators, as they provide structure and simplify the analysis of transformation behaviors, especially in the context of symmetric matrices and their orthogonal complements.

Orthogonal Complement

The orthogonal complement of a subspace in an inner product space consists of all vectors that are orthogonal to every vector in that subspace. In linear algebra, this concept helps in decomposing vector spaces into simpler, non-overlapping parts, which is particularly useful when examining how matrices act on different parts of the space and verifying the invariance under certain transformations.

Eigenvector

An eigenvector of a matrix is a nonzero vector that, when the matrix is applied to it, results only in a scalar multiple of itself. This scalar multiple is known as the eigenvalue. In the context of symmetric matrices, eigenvectors are especially important because they can always be chosen to be orthogonal, ensuring that the space can be decomposed into invariant subspaces under the transformation represented by the matrix.

Symmetric Matrix

A symmetric matrix is a square matrix that is equal to its transpose. This property implies that all its eigenvalues are real and that eigenvectors corresponding to different eigenvalues are orthogonal. Due to these features, symmetric matrices play a central role in many areas of linear algebra, particularly when exploring invariant subspaces and orthogonality properties of eigenvectors.

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