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Eigenvalues of real part of positive definite hermitian matrix

Matrix elements of an inverse Hermitian matrixTransition Matrix eigenvalues constraintsEigenvalues of a matrix of complex numbersProving if A is an Hermitian matrix with nonnegative eigenvalues, A is positive semidefinite.Decide the range of eigenvalues for $A+B$Signs of real part eigenvalues for nonsymmetric matrix.The relationship between diagonal entries and eigenvalues of a diagonalizable matrixRelation Between Eigenvalues of a Matrix and its Real PartGeneralized eigenvalue problem of Hermitian matrix (exist complex eigenvalues)Transforming a non hermitian matrix into a hermitian one

1

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Lets say we have the $ntimes n$ positive definite hermitian matrix $mathbf{A}$. Is there any clear relation between the Eigenvalues of $mathbf{A}$ and the Eigenvalues of its real part $mathbf{B}=Re {mathbf{A}}$?

The only (weak) relation I can think of can be found with the help of Gersgorins theorem, namely, we know that all Eigenvalues of $mathbf{A}$ lie within the Gersgorin disks in the complex plane whose centers are given by the diagonal elements of $mathbf{A}$ and whose radii are the sum of the respective absolute off diagonal entries of either only columns or only rows. Due to the matrix being hermitian these radii are equal for both rows and columns. Furthermore, our Eigenvalues are constraint to only be on the real axis within all circles.

Since we now want to find the Eigenvalues of the matrix real part $mathbf{B}$, we know that these Eigenvalues lie at least within the same lines. Moreover, as we reduce the respective circles' radii by taking the real part, we actually know that area in which the eigenvalues generally could be is also reduced (e.g. tightened to the centers).

But that's about all I can think of and it might be that I'm missing something obvious, maybe even a direct relation. I am especially interested in how the smallest eigenvalues of both matrices compare (and maybe a bound on those). I would thus be very glad for any kind of hints/help!

linear-algebra matrices eigenvalues-eigenvectors

share|cite|improve this question

edited Mar 19 at 15:33

J. Doe

asked Mar 19 at 15:26

J. DoeJ. Doe

104

$endgroup$

add a comment | 

1

$begingroup$

Lets say we have the $ntimes n$ positive definite hermitian matrix $mathbf{A}$. Is there any clear relation between the Eigenvalues of $mathbf{A}$ and the Eigenvalues of its real part $mathbf{B}=Re {mathbf{A}}$?

The only (weak) relation I can think of can be found with the help of Gersgorins theorem, namely, we know that all Eigenvalues of $mathbf{A}$ lie within the Gersgorin disks in the complex plane whose centers are given by the diagonal elements of $mathbf{A}$ and whose radii are the sum of the respective absolute off diagonal entries of either only columns or only rows. Due to the matrix being hermitian these radii are equal for both rows and columns. Furthermore, our Eigenvalues are constraint to only be on the real axis within all circles.

Since we now want to find the Eigenvalues of the matrix real part $mathbf{B}$, we know that these Eigenvalues lie at least within the same lines. Moreover, as we reduce the respective circles' radii by taking the real part, we actually know that area in which the eigenvalues generally could be is also reduced (e.g. tightened to the centers).

But that's about all I can think of and it might be that I'm missing something obvious, maybe even a direct relation. I am especially interested in how the smallest eigenvalues of both matrices compare (and maybe a bound on those). I would thus be very glad for any kind of hints/help!

linear-algebra matrices eigenvalues-eigenvectors

share|cite|improve this question

edited Mar 19 at 15:33

J. Doe

asked Mar 19 at 15:26

J. DoeJ. Doe

104

$endgroup$

add a comment | 

$begingroup$

Lets say we have the $ntimes n$ positive definite hermitian matrix $mathbf{A}$. Is there any clear relation between the Eigenvalues of $mathbf{A}$ and the Eigenvalues of its real part $mathbf{B}=Re {mathbf{A}}$?

The only (weak) relation I can think of can be found with the help of Gersgorins theorem, namely, we know that all Eigenvalues of $mathbf{A}$ lie within the Gersgorin disks in the complex plane whose centers are given by the diagonal elements of $mathbf{A}$ and whose radii are the sum of the respective absolute off diagonal entries of either only columns or only rows. Due to the matrix being hermitian these radii are equal for both rows and columns. Furthermore, our Eigenvalues are constraint to only be on the real axis within all circles.

Since we now want to find the Eigenvalues of the matrix real part $mathbf{B}$, we know that these Eigenvalues lie at least within the same lines. Moreover, as we reduce the respective circles' radii by taking the real part, we actually know that area in which the eigenvalues generally could be is also reduced (e.g. tightened to the centers).

But that's about all I can think of and it might be that I'm missing something obvious, maybe even a direct relation. I am especially interested in how the smallest eigenvalues of both matrices compare (and maybe a bound on those). I would thus be very glad for any kind of hints/help!

linear-algebra matrices eigenvalues-eigenvectors

share|cite|improve this question

edited Mar 19 at 15:33

J. Doe

asked Mar 19 at 15:26

J. DoeJ. Doe

104

$endgroup$

share|cite|improve this question

edited Mar 19 at 15:33

J. Doe

asked Mar 19 at 15:26

J. DoeJ. Doe

104

share|cite|improve this question

edited Mar 19 at 15:33

J. Doe

asked Mar 19 at 15:26

J. DoeJ. Doe

104

asked Mar 19 at 15:26

J. DoeJ. Doe

104

asked Mar 19 at 15:26

J. DoeJ. Doe

104