Let 𝐮1, …, 𝐮n be an orthonormal basis of R^n. Prove that it forms an eigenvector basis for some

step 1 Okay, this question gives us an orthogonal matrix, and it wants us to prove that the columns of

Let 𝐮1, …, 𝐮n be an orthonormal basis of R^n. Prove that it...

Let $\mathbf{u}_1, \ldots, \mathbf{u}_n$ be an orthonormal basis of $\mathrm{R}^n$. Prove that it forms an eigenvector basis for some symmetric $n \times n$ matrix $A$. Can you characterize all such matrices?

Key Concepts

Spectral Theorem

The spectral theorem states that any symmetric matrix can be diagonalized by an orthogonal matrix. In other words, for any symmetric matrix A, there exists an orthogonal matrix P (whose columns are eigenvectors of A) and a diagonal matrix D (whose entries are the eigenvalues of A) such that A = PDP^T. This theorem is fundamental in understanding the structure of symmetric matrices and guarantees the existence of an orthonormal eigenvector basis.

Orthogonal Diagonalization

Orthogonal diagonalization is the process of expressing a symmetric matrix in the form A = PDP^T, where P is an orthogonal matrix and D is a diagonal matrix. This process relies on the fact that symmetric matrices have real eigenvalues and that the eigenvectors corresponding to distinct eigenvalues are orthogonal. This property is pivotal in many applications as it simplifies matrix computations significantly.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are intrinsic to the study of linear transformations represented by matrices. For a given matrix A, an eigenvector is a nonzero vector that only gets scaled (by an eigenvalue) when A is applied to it. In the context of symmetric matrices, all eigenvalues are real and there exists a complete set of mutually orthogonal eigenvectors, forming a basis for the vector space.

Symmetric Matrices

A symmetric matrix is one that is equal to its transpose, meaning A = A^T. This property ensures that symmetric matrices have a number of desirable characteristics, including real eigenvalues and the existence of an orthonormal eigenvector basis. These features make symmetric matrices especially important in various fields such as physics, statistics, and numerical analysis.

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