Questions tagged [hermitian-matrices]
A Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose.
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Prove that if $A$ is hermitian then $\|A\|=\max_{u^\dagger u=1}|u^\dagger Au|=|\lambda_i|_\max$
Prove that when $A$ is hermitian then
$$\|A\|=\max_{u^\dagger u=1}|u^\dagger Au|=\lambda$$
where $\lambda$ is the maximum of the set $|\lambda_i|$ and $\lambda_i$ are the eigenvalues of the matrix $A$....
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Does the arbitrary polynomials of the N-dimensional irreducible representation of SU(2) generate U(N)?
Assuming I have the $N$-dimensional irreducible representation of $SU(2)$ (The $N$-dimensional spin matrices $Sx$,$Sy$ and $Sz$) and I can consruct arbitrary polynomials from them of the form:
$$\sum{...
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Are these conditions sufficient to guarantee positive definiteness?
Let $f: \mathbb{C}^2 \to \mathbb{C}^n$ be the map
$$ f(u, v) = (u^{n-1}, u^{n-2}v, \ldots, u^{n-k} v^{k-1}, \ldots, v^{n-1})^T$$
so that $f$ is the rational normal curve in homogeneous coordinates.
...
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Singular Value Decomposition(SVD) of sum of two Hermitian matrices.
Suppose the Singular Value Decomposition(SVD) of an $m \times n$ matrix $A=U_1S_1V_1^H$ where $V_1^H$ denotes the Hermitian conjugate of $V_1$. Then the SVD of $AA^T=U_1S_1V_1^HV_1S_1^HU_1^H=U_1S_1^...
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Semidefinite programming: flat directions and "not numerically HPD"
My question involves semidefinite programming (SDP) in the sense of attempting to find some vector $\alpha^{\mu}$ that satisfies the following conditions:
Normalisation: $\alpha^{\mu}n_{\mu} = 1$
...
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Eigenvalues of sums of almost commuting Hermitian matrices
I am considering $n \times n$ Hermitian matrices $A, B,$ and $C$ such that $A + B = C$ and with eigenvalues $a_{i}$, $b_{i}$, $c_{i}$ ordered so that $a_{1} \geq a_{2} \geq \cdots \geq a_{n}$ etc... I ...
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Which matrices can be square roots of Hermitian matrices?
I have a square matrix $Q$. (I can additionally assume that it is invertible
${\rm det}\,Q \ne 0$,
but I don't see how this helps.) I define yet another square matrix
$Q_2 = Q^2$.
This new matrix is ...
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Eigenvalues of a matrix $A$ multiplied by its Hermitian transpose $A^HA$ is the square of the original eigenvalues? [duplicate]
Say we have a matrix $A$.
We know the second-induced norm of it will be $||A||_2 = \sqrt{\lambda_i}$ where $\lambda_i$ is the max eigenvalues of matrix $A^{H}A$
If $A$ is a normal matrix, how do we ...
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PSD Hermitian Matrix made up of PSD Hermitian Matrices property
In my research on correlation functions of partially coherent light, a $2\times 2$ matrix of the following form appears:
$$
W=\begin{pmatrix}
W_{xx} & W_{xy} \\
W_{yx} & W_{yy}
\end{pmatrix}$...
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Let $A$ any matrix so $A^H A$ is Hermitian and $A^H A $ is positive definite.
Question:
Let $A$ any matrix so $A^H A$ is Hermitian and $A^H A $ is positive definite.
Answer:
The first part is easy as: $A^H A =A A^H \Rightarrow AA^H = (A^HA)^H=(AA^H)^H$.
But how to prove (...
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Is there any condition for a square matrix that guarantees it has at least 1 positive *and* at least 1 negative eigenvalue?
Wondering if there exists any known condition for a square matrix that guarantees it has at least 1 positive and at least 1 negative eigenvalue?
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Proving not all generators of the Lorentz algebra are (anti)-hermitian
Suppose we have the Lorentz algebra $\mathfrak{so}(1,3)$ with the generators $M_{\mu\nu} = - M_{\nu\mu}$ and commutators between them are $$[M_{\mu\nu}, M_{\sigma\rho}] = g_{\nu\sigma}M_{\mu\rho} - g_{...
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Show that $H_c = (cI - H)(I-\bar{c}H)^{-1}$ is unitary if H is unitary.
I'm struggling with the following question.
Let H be a Unitary matrix and define $H_c = (cI - H)(I-\bar{c}H)^{-1}$ for $c \in \mathbb{C}$. Show that $H_c$ is also unitary.
Using properties of the ...
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Canonical form of a hermitian matrix with respect to congruence
Just a little warning: my english is not that good so the text below can be really confusing.
How can I proof that every hermitian matrix, $H\in C^{n\times n}$, is congruent to a matrix:
\begin{align}
...
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Do I have to study Complex Analysis before Hermitian Matrix?
When I taught myself linear algebra using some textbooks I found that Hermitian Matrix has been mentioned quite a few times; and I see Hermitian Matrix has applications in statistics (time series ...
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Decomposition of hermitian matrix as difference of positive semidefinite matrices
In my reference, Box 11.2, Page 512, Chapter 11, Entropy and Information, Quantum Computation and Quantum Information by Nielsen and Chuang, proof of the Fannes' inequality contains
here $\rho$ and $\...
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numerical radius
Suppose $A$ is a $n \times n$ complex matrix and there exists a hermitian matrix X such that
$$\begin{pmatrix}
I + X & A \\
A^* & I-X
\end{pmatrix} \geq 0$$
Prove that for every $y \in \mathbb{...
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Why do eigenvalues of Hermitian/Unitary maps have to be distinct for the eigenvectors to be orthogonal?
Some things I know:
Eigenvectors corresponding to DISTINCT eigenvalues are orthogonal for Hermitian and Unitary maps.
Hermitian and Unitary maps are normal.
Normal maps have an orthogonal basis of ...
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Are all positive operators Hermitian?
A special subclass of Hermitian operators is extremely important. This is the positive
operators. A positive operator A is defined to be an operator such that for any vector $|v\rangle$,
$(|v\rangle, ...
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For projection matrices, which vector of the outer product should be complex conjugated?
I have a conjugation wrong somewhere in my definitions, and I can't work out where it is. I want to define the standard matrix for a projection operator. If you can provide correct and standard ...
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Show that $\lambda_{2}(zz^\ast)=\lambda_{n-1}(zz^\ast)=0$.
Show that $\lambda_{2}(zz^\ast)=\lambda_{n-1}(zz^\ast)=0$ where $z\in \mathbf{C}^n$ and $n\geq 2$.
I know that $zz^\ast$ is a Hermitian matrix, and all the eigenvalues are nonnegative, but how to ...
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If $A^*= A$ and $A^m= 0$, then $A= 0$.
If $A^*= A$ and $A^m= 0$, then $A= 0$.
My attempt:
If $m = 2$, then $\text{tr}(A^*A)= 0 \Rightarrow A=0$
and then the result follows for $m = \{2,4,8,16,32,\ldots\}$.
But I don't know how it works in ...
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Showing that AA$^*$ and A$^*$A are positive semi-definite and that they have the same nonnegative eigenvalues
Let A be an m × n complex matrix. I am trying to show that AA$^*$ and A$^*$A are positive semi-definite, (P.D.S), and they have the same nonnegative eigenvalues.
I have started by proving that AA$^*$ ...
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If the matrix exponential is unitary, is the exponent necessarily skew-Hermitian?
If $A$ is skew-Hermitian, then $e^A$ is unitary. But is the converse true?
That is, if $e^A$ is unitary, is $A$ necessarily skew-Hermitian?
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Let $\hat{A}=\left[\begin{array}{cc}A&y\\y^\ast&a\end{array}\right]$. what's the possible value of rank $\hat{A}$-rank $A$?
If $A\in M_n$ is Hermitian, $a\in \mathbf{R}$, and $y\in \mathbf{C}^n$.
Let $\hat{A}=\left[\begin{array}{cc}A&y\\y^\ast&a\end{array}\right]$. Then what's the possible value of rank $\hat{A}$-...
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If A,B are hermitian matrices and all eigenvalues of B are positive, then why all eigenvalues of AB are real?
If A, and B are hermitian matrices and all eigenvalues of B are positive, then why eigenvalues of AB is real?
I can write $x^\ast Bx>0$, and $x^\ast Ax\in \mathbb{R}$, and then how about $x^\ast ...
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Show that $|x^\ast Ay|=|x^\ast By|$ iff. $A=e^{i\theta}B$.
Let $A,B\in M_n$. Show that $|x^\ast Ay|=|x^\ast By|$ for all $x,y\in \mathbb{C}^n$ iff $A=e^{i\theta}B$ for some $\theta\in \mathbb{R}$.
The one side is easy to prove by plugging $e^{i\theta}B$ in ...
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Eigenvalues of a block Hermitian matrix
Suppose I have a matrix like the following
$$A= \begin{pmatrix}
B & y \\
y^{*} & a \\
\end{pmatrix}$$
where $B$ is a Hermitian matrix. How can I prove that
\begin{equation}
\lambda_{1}(A)\leq \...
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Derivative of Hermitian sesquilinear form with respect to its own matrix
Let $H$ be an $n \times n$ Hermitian matrix (in my work, it's also positive semidefinite, if that makes a difference) and $a,b \in \mathbb{C}^n$, with $\lambda(H) = \langle a \vert H \vert b \rangle$. ...
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Can every non-hermitian matrix be written as a sum of hermitan matrices?
If $X$ is non-hermitian i.e., $X^\dagger = (X^*)^T \ne X$, can one express it as $X = \sum_i \alpha_i Y_i$ where $\alpha_i$ is a complex scalar and $Y_i$ is hermitian?
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Expectation of $e^{i\alpha H}$
I have $3\times 3$ a matrix $A$ defined as
$$A=e^{i\alpha H}, $$
with $H$ a $3\times 3$ random Hermitian matrix, and $\alpha\in[0,\infty]$. I am trying to determine two things: Can we say anything on ...
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What is the matrix representing a positive operator called?
Considering finite dimensional spaces: Positive operators are a subset of hermitian operators. They have a matrix representation. Therefore one often uses both terms (the operator and the matrix ...
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Diagonalizing a Hermitian Matrix
I have been given a $3×3$ Hermitian matrix $H$. I am asked to come up with a non singular matrix $P$ such that $D=P^{T}H\overline{P}$ where $P^{T}$ represents transpose of matrix $P$ while $\overline{...
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Given two unitary and hermitian matrices with equal traces, Prove that they are similar and congruence matrices.
Given two unitary and hermitian matrices with equal traces $A,B$, I'm trying to prove that they are similar and congruence matrices.
I proved previously that if these two matrices have equal ...
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Can we always find this kind of unitary matrix to diagonalize this special form hermitian matrix?
Suppose we have a hermitian matrix $M$ of form $$\left( \begin{matrix}
\alpha& -\beta ^*\\
\beta& -\alpha ^*\\
\end{matrix} \right) $$ where $\alpha$ is a hermitian matrix and $\beta$ is ...
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Simultaneous diagonalization of $\frac{A+A^{*}}{2}$ and $(AA^{*})^{\frac{1}{2}}$
CONVEX AND CONCAVE FUNCTIONS OF SINGULAR
VALUES OF MATRIX SUMS
This question comes from the proof of theorem $2$ linked above.
The author firstly defines $|A|=(AA^{*})^\frac{1}{2}.$
If $H $ and $K$ ...
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Relationship between the diagonal elements of a Hermitian matrix and its eigenvalues
CONVEX AND CONCAVE FUNCTIONS OF SINGULAR
VALUES OF MATRIX SUMS
This question comes from the proof of theorem $1$ linked above.
Assume $A,B,C=A+B$ are $n\times n$ Hermitian positive semidefinite ...
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Non-Hermitian, non-unitary square matrix factorized into unitary and Hermitian matrices
Is it (always) possible to write a non-Hermitian, non-unitary square matrix as a product of A and B such that A is unitary (but may or may not be Hermitian) and B is Hermitian (but may or may not be ...
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Relations between eigenvalues and eigenvectors of Hermitian matrices $A$ and $A^2$
What are the relations between the eigenvalues and eigenvectors of a Hermitian matrix $A$ and its square $A^2$?
I do know that if $(\lambda, v)$ is an eigenpair of $A$, then ($\lambda^2$, $v$) is an ...
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The proof of $\langle Ax,x\rangle=\langle x,Ax\rangle$ for all $x\in \mathbb C^n$ $\Rightarrow$ $A$ is Hermitian.
Let $\langle \cdot, \cdot \rangle$ be Hermitian inner product and $A=(a_{ij})_{1\leqq i,j\leqq n}$ be $n\times n$ matrix of $\mathbb C.$
Prove this claim.
Claim :
If $\langle Ax,x\rangle=\langle x,Ax\...
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$A$ is Hermitian and $(Ax,x)=0 \Rightarrow x=0$.
Let $A$ be $n\times n$ Hermitian matrix, and $(\cdot ,\cdot)$ be Hermitian inner product.
Define the condition (i) as :
(i) $(Ax,x)=0 \Rightarrow x=0$.
Prove that
if $A$ satisfies (i), then $A$ is ...
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Acting the derivative operator on a bra to its left
Let $f(x)$ be a ket, and $\langle f(x)|$ be the corresponding bra.
Start with
$$\frac{d}{dx} |f(x)\rangle = |f'(x)\rangle$$
Take transpose conjugate of both sides. Since derivative is anti Hermitian, ...
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Hermitian adjoint - can I apply star to eigenvalue? $T(v)=kv$ $ >> $ $T^*(v)=k^*v$
Is what I wrote true? can I apply Star ( $^*$) like that and get the equality?
Sorry if you can find online what I said ( my native is not English and I didn't know how to write - as you see I wrote ...
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Eigenvalues of block matrix with zero diagonal
I'm looking for the eigenvalues of a block matrix of the form
$$
M=\begin{pmatrix}
\boldsymbol{0} & C \\
C^\dagger & \boldsymbol{0}
\end{pmatrix}=
\begin{pmatrix}
\boldsymbol{0} & H(I-uu^\...
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Express non-square matrix with orthonormal columns as function of anti-Hermitian matrix?
For unitary matrices, it is known that for any $U$, there exists a Hermitian matrix $H$ such that
$$U=e^{iH}$$
and similarly, for any $H$ which is Hermitian, $e^{iH}$ is unitary.
Furthermore, for ...
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How to rigorously show that the maximum variance of an hermitian matrix is $\left( \frac{h_{max}-h_{min}}{2} \right) ^2$?
Suppose we have an hermitian matrix $H$ with eigenvalues $h_i$, and I want to get the minimum value of its variance, i.e. $\vec{v}^{\dagger}H^2\vec{v}-\left( \vec{v}^{\dagger}H\vec{v} \right) ^2$. How ...
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What's the difference of the pde technique between solving pde on riemannian manifold and on hermitian manifold?
I'm new to complex geometry.
I find that for some equations, I can easily get the existence of the solution on riemannian manifold, but when they are on hermitian manifold, I don't even know how to ...
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Real-symmetric vs Hermitian Matrices
Let $B \in \mathbb{C}^{n \times n}$ be a Hermitian matrix, $B^{R}$ and $B^{I}$ its real and imaginary parts, respectively. It can be converted to a real-symmetric matrix via this construction.
$$
\...
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Can we always let two uncommute hermitian matrices $A$ and $B$ commute by enlarge the dimension?
For example, we have two hermitian matrices $$A=\left( \begin{matrix}
1& 0\\
0& -1\\
\end{matrix} \right) ,B=\left( \begin{matrix}
0& 1\\
1& 0\\
\end{matrix} \right) .$$
And ...
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Skew-hermitian also a contraction?
If a matrix is skew-hermitian, is it also a contraction?
I've been playing around with skew-hermitian operators on Hilbert spaces, and found this to be true on a couple of examples. So I was wondering ...
