What is the geometrical significance of a complex-valued singular value decomposition?
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Suppose you have a 2x2 real-valued matrix, $mathbf{A}$. If you perform a singular value decomposition (SVD), then this can be understood geometrically as a decomposition of $mathbf{A}$ into a 2-D rotation, scaling and second 2-D rotation of the form:
$$mathbf{A} = mathbf{R_1 S R_2}$$
However, if the entries of $mathbf{A}$ are complex, then what is the geometric meaning of this? Now $mathbf{R_1}$ and $mathbf{R_2}$ will also be complex. Are these still rotation matrices? Is there an additional underlying rotation happening in the complex plane? What does it mean if you have a "complex rotation matrix"?
Any help, clarification or references is appreciated.
linear-algebra complex-numbers linear-transformations svd
asked Nov 15 at 20:42
Darcy
14411
add a comment |
up vote
0
down vote
favorite
Suppose you have a 2x2 real-valued matrix, $mathbf{A}$. If you perform a singular value decomposition (SVD), then this can be understood geometrically as a decomposition of $mathbf{A}$ into a 2-D rotation, scaling and second 2-D rotation of the form:
$$mathbf{A} = mathbf{R_1 S R_2}$$
However, if the entries of $mathbf{A}$ are complex, then what is the geometric meaning of this? Now $mathbf{R_1}$ and $mathbf{R_2}$ will also be complex. Are these still rotation matrices? Is there an additional underlying rotation happening in the complex plane? What does it mean if you have a "complex rotation matrix"?
Any help, clarification or references is appreciated.
linear-algebra complex-numbers linear-transformations svd
asked Nov 15 at 20:42
Darcy
14411
They'll be unitary matrices, which can still be thought of as rotations in a suitable sense.
– Qiaochu Yuan
Nov 15 at 21:27
Rotation through what angle though? What does it mean to have a "complex rotation matrix"? The matrix multiplication does not preserve real and imaginary components as separate. For example, if $mathbf{R_1}$ is a complex rotation matrix and you take some real valued vector, $mathbf{v}$, then $mathbf{R_1v}$ will be complex valued as well. I'm just struggling to understand what this actually means geometrically.
– Darcy
Nov 15 at 21:59
The eigenvalues of a unitary matrix, like an orthogonal matrix, are unit complex numbers $e^{i theta}$, so can be thought of in terms of rotations by $theta$.
– Qiaochu Yuan
Nov 15 at 22:02
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Suppose you have a 2x2 real-valued matrix, $mathbf{A}$. If you perform a singular value decomposition (SVD), then this can be understood geometrically as a decomposition of $mathbf{A}$ into a 2-D rotation, scaling and second 2-D rotation of the form:
$$mathbf{A} = mathbf{R_1 S R_2}$$
However, if the entries of $mathbf{A}$ are complex, then what is the geometric meaning of this? Now $mathbf{R_1}$ and $mathbf{R_2}$ will also be complex. Are these still rotation matrices? Is there an additional underlying rotation happening in the complex plane? What does it mean if you have a "complex rotation matrix"?
Any help, clarification or references is appreciated.
linear-algebra complex-numbers linear-transformations svd
asked Nov 15 at 20:42
Darcy
14411
Suppose you have a 2x2 real-valued matrix, $mathbf{A}$. If you perform a singular value decomposition (SVD), then this can be understood geometrically as a decomposition of $mathbf{A}$ into a 2-D rotation, scaling and second 2-D rotation of the form:
$$mathbf{A} = mathbf{R_1 S R_2}$$
However, if the entries of $mathbf{A}$ are complex, then what is the geometric meaning of this? Now $mathbf{R_1}$ and $mathbf{R_2}$ will also be complex. Are these still rotation matrices? Is there an additional underlying rotation happening in the complex plane? What does it mean if you have a "complex rotation matrix"?
Any help, clarification or references is appreciated.
linear-algebra complex-numbers linear-transformations svd
linear-algebra complex-numbers linear-transformations svd
asked Nov 15 at 20:42
Darcy
14411
asked Nov 15 at 20:42
Darcy
14411
asked Nov 15 at 20:42
Darcy
14411
asked Nov 15 at 20:42
Darcy
14411
asked Nov 15 at 20:42
Darcy
14411
14411
They'll be unitary matrices, which can still be thought of as rotations in a suitable sense.
– Qiaochu Yuan
Nov 15 at 21:27
Rotation through what angle though? What does it mean to have a "complex rotation matrix"? The matrix multiplication does not preserve real and imaginary components as separate. For example, if $mathbf{R_1}$ is a complex rotation matrix and you take some real valued vector, $mathbf{v}$, then $mathbf{R_1v}$ will be complex valued as well. I'm just struggling to understand what this actually means geometrically.
– Darcy
Nov 15 at 21:59
The eigenvalues of a unitary matrix, like an orthogonal matrix, are unit complex numbers $e^{i theta}$, so can be thought of in terms of rotations by $theta$.
– Qiaochu Yuan
Nov 15 at 22:02
add a comment |
They'll be unitary matrices, which can still be thought of as rotations in a suitable sense.
– Qiaochu Yuan
Nov 15 at 21:27
Rotation through what angle though? What does it mean to have a "complex rotation matrix"? The matrix multiplication does not preserve real and imaginary components as separate. For example, if $mathbf{R_1}$ is a complex rotation matrix and you take some real valued vector, $mathbf{v}$, then $mathbf{R_1v}$ will be complex valued as well. I'm just struggling to understand what this actually means geometrically.
– Darcy
Nov 15 at 21:59
The eigenvalues of a unitary matrix, like an orthogonal matrix, are unit complex numbers $e^{i theta}$, so can be thought of in terms of rotations by $theta$.
– Qiaochu Yuan
Nov 15 at 22:02
They'll be unitary matrices, which can still be thought of as rotations in a suitable sense.
– Qiaochu Yuan
Nov 15 at 21:27
They'll be unitary matrices, which can still be thought of as rotations in a suitable sense.
– Qiaochu Yuan
Nov 15 at 21:27
Rotation through what angle though? What does it mean to have a "complex rotation matrix"? The matrix multiplication does not preserve real and imaginary components as separate. For example, if $mathbf{R_1}$ is a complex rotation matrix and you take some real valued vector, $mathbf{v}$, then $mathbf{R_1v}$ will be complex valued as well. I'm just struggling to understand what this actually means geometrically.
– Darcy
Nov 15 at 21:59
Rotation through what angle though? What does it mean to have a "complex rotation matrix"? The matrix multiplication does not preserve real and imaginary components as separate. For example, if $mathbf{R_1}$ is a complex rotation matrix and you take some real valued vector, $mathbf{v}$, then $mathbf{R_1v}$ will be complex valued as well. I'm just struggling to understand what this actually means geometrically.
– Darcy
Nov 15 at 21:59
The eigenvalues of a unitary matrix, like an orthogonal matrix, are unit complex numbers $e^{i theta}$, so can be thought of in terms of rotations by $theta$.
– Qiaochu Yuan
Nov 15 at 22:02
The eigenvalues of a unitary matrix, like an orthogonal matrix, are unit complex numbers $e^{i theta}$, so can be thought of in terms of rotations by $theta$.
– Qiaochu Yuan
Nov 15 at 22:02
add a comment |
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They'll be unitary matrices, which can still be thought of as rotations in a suitable sense.
– Qiaochu Yuan
Nov 15 at 21:27
Rotation through what angle though? What does it mean to have a "complex rotation matrix"? The matrix multiplication does not preserve real and imaginary components as separate. For example, if $mathbf{R_1}$ is a complex rotation matrix and you take some real valued vector, $mathbf{v}$, then $mathbf{R_1v}$ will be complex valued as well. I'm just struggling to understand what this actually means geometrically.
– Darcy
Nov 15 at 21:59
The eigenvalues of a unitary matrix, like an orthogonal matrix, are unit complex numbers $e^{i theta}$, so can be thought of in terms of rotations by $theta$.
– Qiaochu Yuan
Nov 15 at 22:02