Prove that a symmetric matrix is negative definite if and only if all its eigenvalues are negati

Step 1: Define negative definite matrix and eigenvalues.u000AA symmetric matrix u005C( A u005C) is said to b

Prove that a symmetric matrix is negative definite if and...

Key Concepts

Negative Definiteness

A matrix is defined to be negative definite if for every nonzero vector x, the quadratic form x?Ax is negative. In the context of symmetric matrices, this is equivalent to all the eigenvalues being negative. This equivalence arises because when the matrix is diagonalized, the quadratic form can be represented as a weighted sum of the eigenvalues, thus reflecting their influence directly on the definiteness of the matrix.

Spectral Theorem

The spectral theorem asserts that every symmetric matrix can be diagonalized by an orthogonal matrix. This means the matrix can be expressed in a form where its eigenvalues appear along the diagonal, simplifying the analysis of its properties by transforming the study of a matrix into the study of its eigenvalues.

Symmetric Matrix

A symmetric matrix is a square matrix that is equal to its own transpose. This property ensures that the matrix has real eigenvalues and orthogonal eigenvectors, making it easier to analyze its behavior in various applications, particularly when studying quadratic forms and optimization problems.

Eigenvalues

Eigenvalues are scalar values associated with a matrix that indicate how the matrix stretches or compresses vectors in particular directions. For symmetric matrices, the eigenvalues are guaranteed to be real, and their signs can provide significant insight into the properties of the matrix, such as stability and definiteness.

Prove that a symmetric matrix is negative definite if and only if all its eigenvalues are negative.

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