# Linear ALgebra: Showing negative definteness

• N00813
In summary, the conversation discusses proving that if M is a real anti-symmetric n x n matrix, then M^2 is a non-positive matrix. The solution involves using the equation x^T M^2 x = (x^T M) (M x) and the fact that anti-symmetry implies M^T = -M. This leads to the conclusion that M^2 is a non-positive matrix.
N00813

## Homework Statement

If M is a real anti-symmetric n x n matrix, M^2 is a real symmetric matrix. Show that M^2 is a non-positive matrix, i.e. x(transposed) M^2 x <= 0, for all vectors x.

## Homework Equations

det(M) = (-1)^n det (M)

## The Attempt at a Solution

I attempted to use the relevant equation above to find the determinant of M^2, and found that it is >=0. Diagonalising M^2 gives me the matrix of diagonal eigenvalues, which shares the same determinant as M^2. Thus, in the eigenvalue basis, I proved y(trans) D y >=0, which is the opposite of what the question wants.

No need whatsoever for determinants or eigenvalues.

Hint: ##x^T M^2 x = (x^T M) (M x) = ?##

1 person
1. Are you sure you didn't mistype that determinant formula?
2. anti-symmetry implies M^T = -M

jbunniii said:
No need whatsoever for determinants or eigenvalues.

Hint: ##x^T M^2 x = (x^T M) (M x) = ?##

= ##(M^T x)^T (Mx) = -(Mx)^T(Mx) = -|Mx|^2 <=0 ##

Thanks.

## 1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of vector spaces, linear transformations, and systems of linear equations. It involves the manipulation and analysis of linear equations and their geometric representations.

## 2. What does it mean for a matrix to be negatively definite?

A matrix is negatively definite if all of its eigenvalues are negative. This means that all of the vectors in the matrix will produce a negative result when multiplied by themselves, and the matrix has a concave shape when graphed.

## 3. How can I determine if a matrix is negatively definite?

To determine if a matrix is negatively definite, you can use the Eigenvalue Decomposition method. This involves finding the eigenvalues of the matrix and checking if they are all negative. Another method is the Cholesky Decomposition method, which involves factoring the matrix into a lower triangular matrix and its transpose.

## 4. What are some applications of negative definiteness in real life?

Negative definiteness has applications in various fields such as physics, economics, and computer science. It is used to analyze the stability of systems, optimize functions, and solve differential equations.

## 5. How can I use negative definiteness to prove a statement or solve a problem?

Negative definiteness can be used to prove statements and solve problems by using the properties of negative definite matrices. For example, you can use the properties of negative definite matrices to prove that a function has a unique minimum value or to determine the stability of a system.

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