How to write matrices as tensors

[JohanL]

I have some simple questions on how to write matrices as tensors.

1.
$$\left(\begin{array}{cc}a_1\\a_2\end{array}\right)+ \left(\begin{array}{cc}b_1\\b_2\end{array}\right)= \left(\begin{array}{cc}c_1\\c_2\end{array}\right)$$

is this equivalent to

$$A^j + B^j = C^j$$

with j = 1,2.

2.

$$1/2\left(\begin{array}{cc}\dot{x}_1 & \dot{x}_2\end{array}\right) \left(\begin{array}{cc}3m & m\\-m & 3m\end{array}\right) \left(\begin{array}{cc}\dot{x}_1 \\\dot{x}_2\end{array}\right)$$

is this equivalent to

$$1/2\dot{x}^{\mu}M_{\mu\nu}\dot{x}^{\nu}$$

$$\mu,\nu=1,2$$

and

$$M_{11}=3m,...,M_{22}=3m$$

 

[dextercioby]

Yes,it's correct in the first case.In the second,u may put it

$$\frac{1}{2}\dot{x}_{\mu}M^{\mu}{}_{\nu} \dot{x}^{\nu}$$

Daniel.

 

[JohanL]

thanks.

Is my 2 wrong?
Im struggling with the contravariant and covariant indicies.
is it because a row vector is a covariant vector and the column vector is a contravariant vector you write it like that...but that can't be right.


another question:
If you have an expression like

$$A^{ijk}B_k$$

i,j,k = 1,2

this is equivalent to 4 expressions

$$A^{111}B_1 + A^{112}B_2$$

$$A^{121}B_1 + A^{122}B_2$$

$$A^{211}B_1 + A^{212}B_2$$

$$A^{221}B_1 + A^{222}B_2$$

 

[dextercioby]

Nope,it's just

$$A^{ij1}B_{1}+A^{ij2}B_{2}$$

,that is a second rank double contravariant tensor with 4 components,the ones you have written.

Daniel.

 

[JohanL]

ok...ty.

2 again.

if you have a tensor

$$1/2\dot{x}^{\mu}M_{\mu\nu}\dot{x}^{\nu}$$

$$\mu,\nu=1,2$$

$$M_{11}=3m,...,M_{22}=3m$$

and write it as a matrix you _dont_ get

$$1/2\left(\begin{array}{cc}\dot{x}_1 & \dot{x}_2\end{array}\right)\left(\begin{array}{cc}3m & m\\-m & 3m\end{array}\right)\left(\begin{array}{cc}\dot{x}_1 \\\dot{x}_2\end{array}\right)$$

?

 

[dextercioby]

Nope.You can't put that expression in matrix form.

Daniel.