True or false: A negative definite matrix must have negative trace and negative determinant.

Name: Problem 18, Chapter 3: Applied Linear Algebra (Undergraduate Texts in Mathematics)
Uploaded: 2020-12-13T20:45:27-08:00
Duration: 1 min 30 s
Channel: Sheryl Ezze
Description: True or false: A negative definite matrix must have negative trace and negative determinant.

step 1 So we want to answer whether the determinant, the question said, the determinant can never be eq

True or false: A negative definite matrix must have negative...

Key Concepts

Determinant

The determinant of a matrix is a scalar value that can be computed as the product of its eigenvalues. For a negative definite matrix, because each eigenvalue is negative, the sign of the determinant depends on the number of dimensions (i.e., the number of eigenvalues). If the matrix is of odd order, the determinant is negative, whereas if it is of even order, the determinant is positive. This nuance is important in understanding that the determinant is not universally negative for all negative definite matrices.

Trace

The trace of a matrix is defined as the sum of its diagonal elements, and equivalently, it is the sum of its eigenvalues. When all the eigenvalues of a matrix are negative, as is the case in a negative definite matrix, the trace will be negative, reflecting the aggregate effect of these negative values.

Negative Definiteness

A matrix is termed negative definite if, for any nonzero vector, the quadratic form associated with the matrix is strictly negative. In terms of eigenvalue analysis, this means that all of the eigenvalues of the matrix are strictly negative, a property that is central to understanding its behavior in quadratic forms and optimization problems.

Eigenvalues

The eigenvalues of a matrix provide critical insight into many of its properties, including definiteness, stability, and transformation behaviors. For negative definite matrices, every eigenvalue is negative, which in turn affects other matrix characteristics such as the trace and determinant.

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