Eigenvalue locations for various matrix types?
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I'm relatively new to linear algebra and so far I'm aware of at least three types of (real valued) matrices where we can get a guarantee about where the eigenvalues live.
Symmetric ($A^T = A $) $lambda in mathbb{R}$
Skew-Symmetric ($A^T = -A $) $lambda in mathbb{C}$ where Re{$lambda$}$=0$
Orthogonal ($Q^TQ=I$) $lambda in mathbb{C}$ where $|lambda|=1$
I'm curious whether there are many other examples of real matrix types (that aren't subsets of the types listed above) where we get similar guarantees about where their eigenvalues sit in the complex plane. (For example, is there a type of real valued matrix that is neither symmetric, skew symmetric, nor orthogonal, but whose eigenvalues can still be proven to lie on the unit circle?) Thanks!
linear-algebra matrices
asked Nov 22 at 19:03
fridayParticle
162
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up vote
3
down vote
favorite
I'm relatively new to linear algebra and so far I'm aware of at least three types of (real valued) matrices where we can get a guarantee about where the eigenvalues live.
Symmetric ($A^T = A $) $lambda in mathbb{R}$
Skew-Symmetric ($A^T = -A $) $lambda in mathbb{C}$ where Re{$lambda$}$=0$
Orthogonal ($Q^TQ=I$) $lambda in mathbb{C}$ where $|lambda|=1$
I'm curious whether there are many other examples of real matrix types (that aren't subsets of the types listed above) where we get similar guarantees about where their eigenvalues sit in the complex plane. (For example, is there a type of real valued matrix that is neither symmetric, skew symmetric, nor orthogonal, but whose eigenvalues can still be proven to lie on the unit circle?) Thanks!
linear-algebra matrices
asked Nov 22 at 19:03
fridayParticle
162
1
You might like Gershgorin circle theorem, and also stochastic matrices: see Proof that the largest eigenvalue of a stochastic matrix is 1. Methods of root location of polynomials apply to companion matrices or any matrix whose characteristic polynomial is known.
– Jean-Claude Arbaut
Nov 22 at 19:11
Thanks! Gershgorin's circle theorem is exactly the sort of "eigenvalue constraining" theorem I was looking for!
– fridayParticle
Nov 22 at 20:53
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
I'm relatively new to linear algebra and so far I'm aware of at least three types of (real valued) matrices where we can get a guarantee about where the eigenvalues live.
Symmetric ($A^T = A $) $lambda in mathbb{R}$
Skew-Symmetric ($A^T = -A $) $lambda in mathbb{C}$ where Re{$lambda$}$=0$
Orthogonal ($Q^TQ=I$) $lambda in mathbb{C}$ where $|lambda|=1$
I'm curious whether there are many other examples of real matrix types (that aren't subsets of the types listed above) where we get similar guarantees about where their eigenvalues sit in the complex plane. (For example, is there a type of real valued matrix that is neither symmetric, skew symmetric, nor orthogonal, but whose eigenvalues can still be proven to lie on the unit circle?) Thanks!
linear-algebra matrices
asked Nov 22 at 19:03
fridayParticle
162
I'm relatively new to linear algebra and so far I'm aware of at least three types of (real valued) matrices where we can get a guarantee about where the eigenvalues live.
Symmetric ($A^T = A $) $lambda in mathbb{R}$
Skew-Symmetric ($A^T = -A $) $lambda in mathbb{C}$ where Re{$lambda$}$=0$
Orthogonal ($Q^TQ=I$) $lambda in mathbb{C}$ where $|lambda|=1$
I'm curious whether there are many other examples of real matrix types (that aren't subsets of the types listed above) where we get similar guarantees about where their eigenvalues sit in the complex plane. (For example, is there a type of real valued matrix that is neither symmetric, skew symmetric, nor orthogonal, but whose eigenvalues can still be proven to lie on the unit circle?) Thanks!
linear-algebra matrices
linear-algebra matrices
asked Nov 22 at 19:03
fridayParticle
162
asked Nov 22 at 19:03
fridayParticle
162
asked Nov 22 at 19:03
fridayParticle
162
asked Nov 22 at 19:03
fridayParticle
162
asked Nov 22 at 19:03
fridayParticle
162
162
1
You might like Gershgorin circle theorem, and also stochastic matrices: see Proof that the largest eigenvalue of a stochastic matrix is 1. Methods of root location of polynomials apply to companion matrices or any matrix whose characteristic polynomial is known.
– Jean-Claude Arbaut
Nov 22 at 19:11
Thanks! Gershgorin's circle theorem is exactly the sort of "eigenvalue constraining" theorem I was looking for!
– fridayParticle
Nov 22 at 20:53
add a comment |
1
You might like Gershgorin circle theorem, and also stochastic matrices: see Proof that the largest eigenvalue of a stochastic matrix is 1. Methods of root location of polynomials apply to companion matrices or any matrix whose characteristic polynomial is known.
– Jean-Claude Arbaut
Nov 22 at 19:11
Thanks! Gershgorin's circle theorem is exactly the sort of "eigenvalue constraining" theorem I was looking for!
– fridayParticle
Nov 22 at 20:53
1
1
You might like Gershgorin circle theorem, and also stochastic matrices: see Proof that the largest eigenvalue of a stochastic matrix is 1. Methods of root location of polynomials apply to companion matrices or any matrix whose characteristic polynomial is known.
– Jean-Claude Arbaut
Nov 22 at 19:11
You might like Gershgorin circle theorem, and also stochastic matrices: see Proof that the largest eigenvalue of a stochastic matrix is 1. Methods of root location of polynomials apply to companion matrices or any matrix whose characteristic polynomial is known.
– Jean-Claude Arbaut
Nov 22 at 19:11
Thanks! Gershgorin's circle theorem is exactly the sort of "eigenvalue constraining" theorem I was looking for!
– fridayParticle
Nov 22 at 20:53
Thanks! Gershgorin's circle theorem is exactly the sort of "eigenvalue constraining" theorem I was looking for!
– fridayParticle
Nov 22 at 20:53
add a comment |
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1
You might like Gershgorin circle theorem, and also stochastic matrices: see Proof that the largest eigenvalue of a stochastic matrix is 1. Methods of root location of polynomials apply to companion matrices or any matrix whose characteristic polynomial is known.
– Jean-Claude Arbaut
Nov 22 at 19:11
Thanks! Gershgorin's circle theorem is exactly the sort of "eigenvalue constraining" theorem I was looking for!
– fridayParticle
Nov 22 at 20:53