MHB Integrating a diagonal 2x2 matrix

AI Thread Summary
The discussion focuses on deriving the expression for \(\mathbf{V}_{-1}^{(D)}\) in the context of integrating a diagonal 2x2 matrix. The equation \(\frac{\partial\mathbf{V}_{-1}^{(D)}}{\partial x} = \frac{i}{2}\begin{bmatrix} -(qr)_t & 0 \\ 0 & (qr)_t \end{bmatrix}\) is rewritten using \(\sigma_3\) to show that it can be expressed in terms of \(\alpha \sigma_3 + c\mathbb{I}\). The relationship \(\alpha_x + \frac{1}{2}i(qr)_t = 0\) indicates how \(\alpha\) is derived from the differential equation. The author concludes that the integration process essentially substitutes one differential equation for another. The thread was later marked as solved by the original poster.
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\(r = r(x, t)\), \(q = q(x, t)\), \(\sigma_3 =
\begin{bmatrix}
1 & 0\\
0 & -1
\end{bmatrix}
\)
I have the equation
\begin{align}
\frac{\partial\mathbf{V}_{-1}^{(D)}}{\partial x} &= \frac{i}{2}\begin{bmatrix}
-(qr)_t & 0\\
0 & (qr)_t
\end{bmatrix}\\
\mathbf{V}_{-1}^{(D)} &= \alpha\sigma_3 + c\mathbb{I}\qquad (*)
\end{align}
where \(\alpha\) is a function of \(x\) and \(t\) related to \(q\) and \(r\) and
\[
\alpha_x + \frac{1}{2}i(qr)_t = 0
\]

How is \((*)\) obtained? I don't see it. I know that \(c\mathbb{I}\) is the matrix constant of integration so I am only focused on how \(\alpha\sigma_3\) comes from integrating the diagonal matrix.

I am trying to figure out the last page of
http://math.arizona.edu/~mcl/Miller/MillerLecture06.pdf
 
Last edited:
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dwsmith said:
\(r = r(x, t)\), \(q = q(x, t)\), \(\sigma_3 =
\begin{bmatrix}
1 & 0\\
0 & -1
\end{bmatrix}
\)
I have the equation
\begin{align}
\frac{\partial\mathbf{V}_{-1}^{(D)}}{\partial x} &= \frac{i}{2}\begin{bmatrix}
-(qr)_t & 0\\
0 & (qr)_t
\end{bmatrix}\\
\mathbf{V}_{-1}^{(D)} &= \alpha\sigma_3 + c\mathbb{I}\qquad (*)
\end{align}
where \(\alpha\) is a function of \(x\) and \(t\) related to \(q\) and \(r\) and
\[
\alpha_x + \frac{1}{2}i(qr)_t = 0
\]

How is \((*)\) obtained? I don't see it. I know that \(c\mathbb{I}\) is the matrix constant of integration so I am only focused on how \(\alpha\sigma_3\) comes from integrating the diagonal matrix.

I am trying to figure out the last page of
http://math.arizona.edu/~mcl/Miller/MillerLecture06.pdf

Hmm. Interesting. Well, we can write
$$\frac{\partial\mathbf{V}_{-1}^{(D)}}{\partial x}=\frac{i}{2}\begin{bmatrix}
-(qr)_t & 0 \\ 0 & (qr)_t \end{bmatrix}=- \frac{i(qr)_{t}}{2} \begin{bmatrix} 1 &0 \\ 0 &-1 \end{bmatrix}= - \frac{i(qr)_{t}}{2} \sigma_{3}.$$
At the very least, if we take
$$\mathbf{V}_{-1}^{(D)} = \alpha \sigma_3 + c\mathbb{I} \qquad (*),$$
where
$$\alpha_x + \frac{1}{2}i(qr)_t = 0,$$
then if we differentiate $(*)$ w.r.t. $x$, we get that
$$\frac{\partial\mathbf{V}_{-1}^{(D)}}{\partial x}= \alpha_{x} \sigma_{3}=
- \frac{i(qr)_{t}}{2} \sigma_{3},$$
which is what we had before. Given that the author has not specified $\alpha$ explicitly, but only given a DE that it satisfies, it looks to me as though he's merely substituted one DE for another.
 
Ackbach said:
Hmm. Interesting. Well, we can write
$$\frac{\partial\mathbf{V}_{-1}^{(D)}}{\partial x}=\frac{i}{2}\begin{bmatrix}
-(qr)_t & 0 \\ 0 & (qr)_t \end{bmatrix}=- \frac{i(qr)_{t}}{2} \begin{bmatrix} 1 &0 \\ 0 &-1 \end{bmatrix}= - \frac{i(qr)_{t}}{2} \sigma_{3}.$$
At the very least, if we take
$$\mathbf{V}_{-1}^{(D)} = \alpha \sigma_3 + c\mathbb{I} \qquad (*),$$
where
$$\alpha_x + \frac{1}{2}i(qr)_t = 0,$$
then if we differentiate $(*)$ w.r.t. $x$, we get that
$$\frac{\partial\mathbf{V}_{-1}^{(D)}}{\partial x}= \alpha_{x} \sigma_{3}=
- \frac{i(qr)_{t}}{2} \sigma_{3},$$
which is what we had before. Given that the author has not specified $\alpha$ explicitly, but only given a DE that it satisfies, it looks to me as though he's merely substituted one DE for another.

Thanks, I actually figured everything out but forgot to mark the thread as solved.
 
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