Coordinates of antisymmetric matrix

In summary, the problem asks for finding the coordinates of the matrix A in terms of the given basis matrices E1, E2, and E3. This involves writing A as a linear combination of E1, E2, and E3, and then solving for the coefficients. This can be done by equating corresponding components of A and the linear combination of E1, E2, and E3, which leads to a system of 3 equations. The solution for the coefficients is (1,-3,0). However, it is possible that the correct solution is (2,-2,1) and further checking may be needed.
  • #1

Homework Statement



Let's say that V is the vector space of all antisymmetric 3x3 matrices. Find the coordinates of the matrix [tex]A=\begin{bmatrix}
0 & 1 & -2\\
-1 & 0 & -3\\
2 & 3 & 0
\end{bmatrix}[/tex] in ratio with the base:

[tex]E_1=\begin{bmatrix}
0 & 1 & 1\\
-1 & 0 & 0\\
-1 & 0 & 0
\end{bmatrix}[/tex]

[tex]E_2=\begin{bmatrix}
0 & 0 & 1\\
0 & 0 & 1\\
-1 & -1 & 0
\end{bmatrix}[/tex]

[tex]E_3=\begin{bmatrix}
0 & -1 & 0\\
1 & 0 & -1\\
0 & 1 & 0
\end{bmatrix}[/tex]

Homework Equations



antisymetric matrix is only if [itex]A^t=-A[/tex]

The Attempt at a Solution



The matrix is equal to:

[tex]f: \mathbb{R}^3 \rightarrow \mathbb{R}^3 , f(x_1,x_2,x_3)=(x_2-2x_3,-x_1-3x_3,2x_1+3x_2)[/tex]

The base is [tex]B={(x_2+x_3,-x_1,-x_1) ; (x_3,x_3,-x_1-x_2) ; (-x_2,x_1-x_3,x_2)}[/tex]

What should I do now?
 
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  • #2
Do you understand what the problem is asking? It's exactly like many other problems you have done in the past: write the given "vector" (the matrix A) as a linear combination of the given basis "vectors" (the matrices E1, E2, E3). That is, find numbers a1, a2, a3 so that A= a1E1+ a2E2+ a3E3. Those numbers are the "coordinates".

Setting corresponding components on both sides equal will give you 9 equations, of course, but they should reduce to 3 independent equations for a1, a2, and a3. For example, the upper left component (A11) of every matrix is 0 so that just becomes 0= 0a1+ 0a2+ 0a3 which is satisfied by all numbers.
 
  • #3
I understand. Thanks for the help, I wasn't sure what the problem was asking for...
I found [tex](a_1,a_2,a_3)=(1,-3,0)[/tex] and in my book get (2,-2,1). Probably is my mistake, I will check again.
 

1. What is an antisymmetric matrix?

An antisymmetric matrix is a square matrix where the values on the main diagonal are all zeros, and the values below the main diagonal are the negative of the values above the main diagonal. This means that the matrix is equal to the negative of its own transpose.

2. How can I identify an antisymmetric matrix?

An antisymmetric matrix can be identified by checking if it is equal to the negative of its own transpose. Additionally, it will have all zeros on the main diagonal and the values below the main diagonal will be the negative of the values above the main diagonal.

3. What are the properties of an antisymmetric matrix?

An antisymmetric matrix has the property that its transpose is equal to the negative of the original matrix. It also has the property that the sum of an antisymmetric matrix and its transpose is always a symmetric matrix.

4. How are antisymmetric matrices used in science?

Antisymmetric matrices are used in various scientific fields, such as physics, engineering, and computer science. They are particularly useful in representing physical quantities that have both magnitude and direction, such as force and velocity.

5. Can an antisymmetric matrix have complex numbers?

Yes, an antisymmetric matrix can have complex numbers as its entries. However, the imaginary parts of the complex numbers must be equal in order for the matrix to remain antisymmetric. This means that the complex numbers must be either purely real or purely imaginary.

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