Show the Polar Factor is the Closest Unitary Matrix Using the Spectral Norm
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For a square matrix $A in mathbb{C}^{n times n}$ with the singular value decomposition $A = USigma V^*$, I want to show that
$$|A - P |_{2} leq |A -W |_{2}$$
Where $P = UV^{*}$ and $W$ is an arbitrary unitary matrix.
It is immediately clear to me that
$$|A - P |_{2} = |USigma V^* - UV^{*}|_2 = |Sigma - I |_2 $$
I also know that the singular values of all unitary matrices are all one. However, I don't know how to combine this fact with properties of the spectral norm to get a proof, if this is indeed the right way.
linear-algebra matrices norm
edited Nov 18 at 0:44
user1551
70.5k566125
asked Nov 17 at 23:49
SimonP
434
add a comment |
up vote
4
down vote
favorite
For a square matrix $A in mathbb{C}^{n times n}$ with the singular value decomposition $A = USigma V^*$, I want to show that
$$|A - P |_{2} leq |A -W |_{2}$$
Where $P = UV^{*}$ and $W$ is an arbitrary unitary matrix.
It is immediately clear to me that
$$|A - P |_{2} = |USigma V^* - UV^{*}|_2 = |Sigma - I |_2 $$
I also know that the singular values of all unitary matrices are all one. However, I don't know how to combine this fact with properties of the spectral norm to get a proof, if this is indeed the right way.
linear-algebra matrices norm
edited Nov 18 at 0:44
user1551
70.5k566125
asked Nov 17 at 23:49
SimonP
434
2
You may find the proof here, Theorem 1.
– A.Γ.
Nov 18 at 0:55
@A.Γ. , Thank you for this.
– SimonP
Nov 28 at 11:34
add a comment |
up vote
4
down vote
favorite
up vote
4
down vote
favorite
For a square matrix $A in mathbb{C}^{n times n}$ with the singular value decomposition $A = USigma V^*$, I want to show that
$$|A - P |_{2} leq |A -W |_{2}$$
Where $P = UV^{*}$ and $W$ is an arbitrary unitary matrix.
It is immediately clear to me that
$$|A - P |_{2} = |USigma V^* - UV^{*}|_2 = |Sigma - I |_2 $$
I also know that the singular values of all unitary matrices are all one. However, I don't know how to combine this fact with properties of the spectral norm to get a proof, if this is indeed the right way.
linear-algebra matrices norm
edited Nov 18 at 0:44
user1551
70.5k566125
asked Nov 17 at 23:49
SimonP
434
For a square matrix $A in mathbb{C}^{n times n}$ with the singular value decomposition $A = USigma V^*$, I want to show that
$$|A - P |_{2} leq |A -W |_{2}$$
Where $P = UV^{*}$ and $W$ is an arbitrary unitary matrix.
It is immediately clear to me that
$$|A - P |_{2} = |USigma V^* - UV^{*}|_2 = |Sigma - I |_2 $$
I also know that the singular values of all unitary matrices are all one. However, I don't know how to combine this fact with properties of the spectral norm to get a proof, if this is indeed the right way.
linear-algebra matrices norm
linear-algebra matrices norm
edited Nov 18 at 0:44
user1551
70.5k566125
asked Nov 17 at 23:49
SimonP
434
edited Nov 18 at 0:44
user1551
70.5k566125
asked Nov 17 at 23:49
SimonP
434
edited Nov 18 at 0:44
user1551
70.5k566125
edited Nov 18 at 0:44
user1551
70.5k566125
edited Nov 18 at 0:44
user1551
70.5k566125
70.5k566125
asked Nov 17 at 23:49
SimonP
434
asked Nov 17 at 23:49
SimonP
434
asked Nov 17 at 23:49
SimonP
434
434
2
You may find the proof here, Theorem 1.
– A.Γ.
Nov 18 at 0:55
@A.Γ. , Thank you for this.
– SimonP
Nov 28 at 11:34
add a comment |
2
You may find the proof here, Theorem 1.
– A.Γ.
Nov 18 at 0:55
@A.Γ. , Thank you for this.
– SimonP
Nov 28 at 11:34
2
2
You may find the proof here, Theorem 1.
– A.Γ.
Nov 18 at 0:55
You may find the proof here, Theorem 1.
– A.Γ.
Nov 18 at 0:55
@A.Γ. , Thank you for this.
– SimonP
Nov 28 at 11:34
@A.Γ. , Thank you for this.
– SimonP
Nov 28 at 11:34
add a comment |
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You may find the proof here, Theorem 1.
– A.Γ.
Nov 18 at 0:55
@A.Γ. , Thank you for this.
– SimonP
Nov 28 at 11:34