# Linear Algebra Homework: Matrix \Gamma^{\mu}p_{\mu} Rank

• Grieverheart
In summary, the conversation discusses a question about the space-time supersymmetry of the superparticle in string theory and how the matrix \Gamma^{\mu}p_{\mu} can have at most half maximal rank when the equations of motion are imposed. The relevant equations are \Gamma^{\mu} p_{\mu} \dot{\theta}=0, where \dot{\theta}(\tau) is an anti-commuting spinor with 32 components in 10 dimensions. The attempt at a solution involves setting up a matrix and using its properties to determine the maximum rank of \Gamma^{\mu}p_{\mu}.
Grieverheart

## Homework Statement

Ok, it seems I need to refresh my linear algebra a bit. In the string theory exams, we had a part about space-time supersymmetry of the superparticle. On of the questions was this:

Argue that the matrix $\Gamma^{\mu}p_{\mu}$ can have at most half maximal rank upon imposing the equations of motion.

(Recall that any matrix can be brought to Jordan normal form, meaning that it is diagonal with at most 1s in places right above the diagonal entries.

## Homework Equations

Equations of Motion (the relevant ones):
$\Gamma^{\mu} p_{\mu} \dot{\theta}=0$, where $\dot{\theta}(\tau)$ is an anti-commuting spinor in space-time and has 32 components (we're working in 10 dimensions.

## The Attempt at a Solution

I write $\Gamma^{\mu}p_{\mu} \dot{\theta}=0$ as (I use 3x3 for simplicity)

$\begin{pmatrix} a & 1 & 0\\ 0 & b & 1\\ 0 & 0 & c \end{pmatrix} \begin{pmatrix} \dot{\theta_0}\\ \dot{\theta_1}\\ \dot{\theta_2} \end{pmatrix}=0$

From this I get 3 equations, but I'm not really sure how they affect the rank.

I think it has something to do with the fact that the equations are linearly independent and so the rank of \Gamma^{\mu}p_{\mu} can be at most 3-2=1, but I'm not sure how to express this mathematically. Any help would be appreciated.

## 1. What is a matrix in linear algebra?

A matrix in linear algebra is a rectangular array of numbers or symbols arranged in rows and columns. It is used to represent linear transformations and solve systems of linear equations.

## 2. What is the significance of the symbol \Gamma^{\mu}p_{\mu} in linear algebra homework?

The symbol \Gamma^{\mu}p_{\mu} represents the dot product of a matrix \Gamma with a vector p in the context of linear algebra. It is commonly used in quantum mechanics to represent the energy-momentum relation of a particle.

## 3. What is the rank of a matrix in linear algebra?

The rank of a matrix in linear algebra is the maximum number of linearly independent rows or columns in the matrix. It is a measure of the "size" of the matrix and is important in determining its properties and solutions to systems of equations.

## 4. How do you calculate the rank of a matrix in linear algebra?

The rank of a matrix can be calculated by performing row and column operations to reduce the matrix to its "echelon form." The number of non-zero rows or columns in the echelon form is equal to the rank of the original matrix.

## 5. Why is understanding matrix rank important in linear algebra?

Understanding matrix rank is important in linear algebra because it allows us to determine the existence and uniqueness of solutions to systems of equations. It also helps us to understand the properties and behavior of linear transformations and their effects on vector spaces.

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