Prove that every symmetric matrix S=K+N can be written as the sum of a positive definite matrix

Prove that every symmetric matrix S=K+N can be written as...

Key Concepts

Spectral Theorem

The spectral theorem states that every symmetric matrix can be diagonalized by an orthogonal matrix, meaning that it can be expressed in terms of its eigenvalues and eigenvectors. This theorem is fundamental because it allows for the decomposition of a matrix into simpler components, which can then be manipulated based on their sign-definiteness.

Symmetric Matrix

A symmetric matrix is one that is equal to its transpose, which guarantees that all of its eigenvalues are real. This property makes symmetric matrices particularly tractable in linear algebra, especially when it comes to their spectral properties and diagonalizability.

Positive Definite Matrix

A positive definite matrix is one for which every nonzero vector results in a strictly positive quadratic form. This concept is central in optimization and numerical analysis because it implies well-behaved energy functions and ensures that local minima are unique.

Negative Definite Matrix

A negative definite matrix is defined similarly to a positive definite matrix, except that its quadratic form is strictly negative for every nonzero vector. This property is often encountered in stability analysis of dynamical systems where it represents a form of 'downward curvature' or concavity.

Matrix Decomposition

Matrix decomposition techniques involve breaking a matrix into a sum or product of matrices with desirable properties. In the context of this problem, the decomposition into positive and negative definite parts leverages the eigenvalue structure provided by the spectral theorem, providing insights into the behavior of the original symmetric matrix.

Transcript

00:01 For this problem, we want to find the symmetric and skew symmetric parts, labeled b and c, of the matrix a equals 492, negative 1, negative 2, 5, 035.

00:10 So, we know that the symmetric part of a can be expressed as a plus a transpose, and the skew symmetric part can be expressed as one -half of a -minus a -transpose.

00:25 So that tells us then that b is going to equal one half of, before i get ahead of myself here, we need to actually write out, well, what's a transpose going to be? a transpose is what happens when we could think about it as flipping it, or swapping the rows and columns, or we could think about it as mirroring it along the diagonal.

00:54 So these would swap, these would swap, and these would swap.

00:58 While the diagonal elements stay in place.

01:02 So we'd have along the diagonal, 4, negative 2, 5, just as we had before.

01:06 Then we'd have negative 1 and 0 down here, and we would have 9 and 3 there, and we'd have 2 and 5 there.

01:22 That gives us a transpose.

01:24 So now looking at what b will be, we just have to do the matrix addition of those two matrices.

01:31 I'm not going to bother writing it all out again.

01:34 We have both the matrices easily in view.

01:38 So i'm just going to do the addition in place.

01:41 So for matrix addition, remember we just add elements, add together the matrices element by element.

01:48 So we'd have 4 plus 4 is going to be 8 in here.

01:51 9 plus negative 1 is going to be 8 in there.