Understanding Symmetric Matrix Properties: A Puzzling Example

[dirk_mec1]

Homework Statement

http://img266.imageshack.us/img266/152/78148531ur5.png

Homework Equations

A is symmetric.

The Attempt at a Solution

First of all if you calculate r^T you'll get q^TA so why it the order reversed in the picture above? Moreover I don't see why it is zero.

 

[quantumdude]

Can you go into more detail as to what those symbols mean? The superscripted T's clearly indicate that some of them are matrices, but are they all matrices? If so, then are they all square? If not, then how many rows and columns does each have?

 

[Architeuthis Dux]

I would like to address the puzzling example and provide a possible explanation for the confusion.

The key to understanding this example lies in the definition of a symmetric matrix. A symmetric matrix is a square matrix that is equal to its transpose, meaning that the elements in the upper triangular portion of the matrix are equal to the elements in the lower triangular portion. In this case, the matrix A is symmetric because it satisfies this definition.

Now, let's take a closer look at the equation rT = qTA. This equation is simply stating that the transpose of r (rT) is equal to the product of q and the transpose of A (qTA). This is a general property of matrices, not just for symmetric matrices.

So why is rT equal to zero in this example? This is because qTA is equal to zero. This may seem puzzling at first, but remember that A is a symmetric matrix. This means that the elements in the upper triangular portion of A are equal to the elements in the lower triangular portion. Therefore, when we take the transpose of A, the elements in the upper triangular portion will become the elements in the lower triangular portion. This means that qTA will have elements that are the same as qTA, but in reverse order. Since the elements in qTA are equal to zero, the elements in the reverse order will also be equal to zero, resulting in a zero matrix.

In summary, the reason why rT is equal to zero in this example is because of the properties of symmetric matrices. It may seem puzzling at first, but once we understand the definition of a symmetric matrix and how it affects the transpose, the solution becomes clear.