What is an affine transformation whose “centered” part is orthogonal called?
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On page 7 of The Meaning of Relativity, Einstein calls transformations of the form
$$x^{overline{i}}=a^{overline{i}}+e_{:i}^{overline{i}}x^{i},$$
or
$$Delta x^{overline{i}}=e_{:i}^{overline{i}}Delta x^{i}$$
"linear orthogonal transformations." The book actually references equation (4), but it's pretty clear that's a typo, with the correct reference being to (3a).
I don't believe it is mathematically correct to call the first transformation linear. I call it an affine transformation.
The second transformation is shown in the text to satisfy
$$e_{:i}^{overline{i}}e_{:j}^{overline{j}}delta_{overline{i}overline{j}}=delta_{ij}.$$
Making it a linear orthogonal transformation. That is, a centered affine transformation which is orthogonal. My question is: what should we call a transformation of the first kind?
linear-transformations definition affine-geometry
asked Nov 26 at 5:58
Steven Hatton
721315
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On page 7 of The Meaning of Relativity, Einstein calls transformations of the form
$$x^{overline{i}}=a^{overline{i}}+e_{:i}^{overline{i}}x^{i},$$
or
$$Delta x^{overline{i}}=e_{:i}^{overline{i}}Delta x^{i}$$
"linear orthogonal transformations." The book actually references equation (4), but it's pretty clear that's a typo, with the correct reference being to (3a).
I don't believe it is mathematically correct to call the first transformation linear. I call it an affine transformation.
The second transformation is shown in the text to satisfy
$$e_{:i}^{overline{i}}e_{:j}^{overline{j}}delta_{overline{i}overline{j}}=delta_{ij}.$$
Making it a linear orthogonal transformation. That is, a centered affine transformation which is orthogonal. My question is: what should we call a transformation of the first kind?
linear-transformations definition affine-geometry
asked Nov 26 at 5:58
Steven Hatton
721315
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
On page 7 of The Meaning of Relativity, Einstein calls transformations of the form
$$x^{overline{i}}=a^{overline{i}}+e_{:i}^{overline{i}}x^{i},$$
or
$$Delta x^{overline{i}}=e_{:i}^{overline{i}}Delta x^{i}$$
"linear orthogonal transformations." The book actually references equation (4), but it's pretty clear that's a typo, with the correct reference being to (3a).
I don't believe it is mathematically correct to call the first transformation linear. I call it an affine transformation.
The second transformation is shown in the text to satisfy
$$e_{:i}^{overline{i}}e_{:j}^{overline{j}}delta_{overline{i}overline{j}}=delta_{ij}.$$
Making it a linear orthogonal transformation. That is, a centered affine transformation which is orthogonal. My question is: what should we call a transformation of the first kind?
linear-transformations definition affine-geometry
asked Nov 26 at 5:58
Steven Hatton
721315
On page 7 of The Meaning of Relativity, Einstein calls transformations of the form
$$x^{overline{i}}=a^{overline{i}}+e_{:i}^{overline{i}}x^{i},$$
or
$$Delta x^{overline{i}}=e_{:i}^{overline{i}}Delta x^{i}$$
"linear orthogonal transformations." The book actually references equation (4), but it's pretty clear that's a typo, with the correct reference being to (3a).
I don't believe it is mathematically correct to call the first transformation linear. I call it an affine transformation.
The second transformation is shown in the text to satisfy
$$e_{:i}^{overline{i}}e_{:j}^{overline{j}}delta_{overline{i}overline{j}}=delta_{ij}.$$
Making it a linear orthogonal transformation. That is, a centered affine transformation which is orthogonal. My question is: what should we call a transformation of the first kind?
linear-transformations definition affine-geometry
linear-transformations definition affine-geometry
asked Nov 26 at 5:58
Steven Hatton
721315
asked Nov 26 at 5:58
Steven Hatton
721315
asked Nov 26 at 5:58
Steven Hatton
721315
asked Nov 26 at 5:58
Steven Hatton
721315
asked Nov 26 at 5:58
Steven Hatton
721315
721315
add a comment |
add a comment |
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