Question

1. (7 marks) Let x and y be 2 × 1 column vectors. Consider the inner product ?x, y?G ? x^T Gy for some symmetric 2 × 2 matrix G = ? e f ? ? f g ?, with det G > 0 and e, g > 0. Find a (nonzero) rotation matrix R_G that satisfies the orthogonality condition ?R_G u, u?G = 0, for all 2 × 1 column vectors u. That is, given the vector u, R_G u produces a 2 × 1 vector that is orthogonal to u with respect to the inner product ?·, ·?G. Your answer will be unique only up to a multiplicative constant. Hint: R_G should be of the form R_G = k ? a b ? ? c -a ?, for some arbitrary (nonzero) constant k. The entries a, b, and c are to be found, and must be independent of u. When solving for a, b, and c, take care to not divide by f, which could possibly be 0. A couple of remarks for your interest - they are not necessary for answering the above question: Remark 1: The constant k may be determined (up to a sign) by imposing the additional condition ||R_G x||_G = ||x||_G, where || · ||_G is defined by ||x||_G ? ??x, x?G (that is, we impose that the rotation preserves the norm). You need not find this constant. Remark 2: When G is the 2 × 2 identity matrix, we of course recover the usual Euclidean inner product and norm. In this case (up to a multiplicative constant), R_G = ? 0 1 ? ? -1 0 ?. Remark 3: Non-Euclidean inner products and norms may be relevant when the space you are considering is not flat. Notice that ?² is a flat surface, so the Euclidean inner product and norm are natural in that setting. If we consider instead a surface that was not flat but had bumps and/or was curved (e.g., the surface of a sphere or torus), the concepts of orthogonality and distance need to be modified; in such cases, the usual Euclidean inner product and norm must be replaced by non-Euclidean ones that reflect properties of the surface.

          1. (7 marks) Let x and y be 2 × 1 column vectors. Consider the inner product ?x, y?G ? x^T Gy for some symmetric 2 × 2 matrix
G = ? e f ?
    ? f g ?,
with det G > 0 and e, g > 0. Find a (nonzero) rotation matrix R_G that satisfies the orthogonality condition
?R_G u, u?G = 0,
for all 2 × 1 column vectors u. That is, given the vector u, R_G u produces a 2 × 1 vector that is orthogonal to u with respect to the inner product ?·, ·?G. Your answer will be unique only up to a multiplicative constant.
Hint: R_G should be of the form
R_G = k ? a b ?
          ? c -a ?,
for some arbitrary (nonzero) constant k. The entries a, b, and c are to be found, and must be independent of u. When solving for a, b, and c, take care to not divide by f, which could possibly be 0.

A couple of remarks for your interest - they are not necessary for answering the above question:

Remark 1: The constant k may be determined (up to a sign) by imposing the additional condition ||R_G x||_G = ||x||_G, where || · ||_G is defined by ||x||_G ? ??x, x?G (that is, we impose that the rotation preserves the norm).
You need not find this constant.

Remark 2: When G is the 2 × 2 identity matrix, we of course recover the usual Euclidean inner product and norm. In this case (up to a multiplicative constant),
R_G = ? 0 1 ?
      ? -1 0 ?.

Remark 3: Non-Euclidean inner products and norms may be relevant when the space you are considering is not flat. Notice that ?² is a flat surface, so the Euclidean inner product and norm are natural in that setting. If we consider instead a surface that was not flat but had bumps and/or was curved (e.g., the surface of a sphere or torus), the concepts of orthogonality and distance need to be modified; in such cases, the usual Euclidean inner product and norm must be replaced by non-Euclidean ones that reflect properties of the surface.

1. (7 marks) Let x and y be 2 × 1 column vectors. Consider the inner product ?x, y?G ? x^T Gy for some symmetric 2 × 2 matrix
G = ? e f ?
? f g ?,
with det G > 0 and e, g > 0. Find a (nonzero) rotation matrix RG that satisfies the orthogonality condition
?RG u, u?G = 0,
for all 2 × 1 column vectors u. That is, given the vector u, RG u produces a 2 × 1 vector that is orthogonal to u with respect to the inner product ?·, ·?G. Your answer will be unique only up to a multiplicative constant.
Hint: RG should be of the form
RG = k ? a b ?
? c -a ?,
for some arbitrary (nonzero) constant k. The entries a, b, and c are to be found, and must be independent of u. When solving for a, b, and c, take care to not divide by f, which could possibly be 0.

A couple of remarks for your interest - they are not necessary for answering the above question:

Remark 1: The constant k may be determined (up to a sign) by imposing the additional condition ||RG x||G = ||x||G, where || · ||G is defined by ||x||G ? ??x, x?G (that is, we impose that the rotation preserves the norm).
You need not find this constant.

Remark 2: When G is the 2 × 2 identity matrix, we of course recover the usual Euclidean inner product and norm. In this case (up to a multiplicative constant),
RG = ? 0 1 ?
? -1 0 ?.

Remark 3: Non-Euclidean inner products and norms may be relevant when the space you are considering is not flat. Notice that ?² is a flat surface, so the Euclidean inner product and norm are natural in that setting. If we consider instead a surface that was not flat but had bumps and/or was curved (e.g., the surface of a sphere or torus), the concepts of orthogonality and distance need to be modified; in such cases, the usual Euclidean inner product and norm must be replaced by non-Euclidean ones that reflect properties of the surface.

Added by Anna A.

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