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Questions tagged [hermitian-matrices]

A Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose.

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An "almost" true inequality for Hermitian matrices

Let $A$ be an $N\times N$ Hermitian matrix. For $p+q$ even, consider the following inequality: $$\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} (A^q)_{ii} \geq \Big(\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} \Big) \Big(\...
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Spectral Theorem for normal maps in complex scalar spaces

I'm revisiting my first year lecture notes, but embarrassingly cannot follow a crucial step in a proof and would like to have some opinions on this: Let $V$ be a finite dimensional complex-scalar ...
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Clifford algebra for hermitian forms

Let $k$ be a CM field with conjugation denoted by $\bar{\cdot}$ and consider an hermitian form $q$ over a $k$ vector space $V$. I want to construct an analogue of what could be the Clifford algebra ...
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Orthogonal projection of a complex valued matrix onto the space of Hermitian matrices

It is well known that any real matrix $A$ can be decomposed as the sum of a symmetric and a skew-symmetric matrix as follows: $$ A= \frac{A+A^T}2+\frac{A-A^T}2. $$ The decomposition is orthogonal, ...
Albert's user avatar
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Infinitesimal rank 1 update to eigendecomposition

Consider a Hermitian matrix $H$ for which we know the decomposition $Q \Lambda Q^\ast$. Let $H'= H + \epsilon~x x^\ast$ for a small $\epsilon$. What is a good way of computing the decomposition of $H'$...
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Best method for sequential small size Hermitian smallest eigenpair problem

I am working on a perhaps rather strange problem. To find the smallest eigenvalue and its eigenvector, for a large number (a few billions) of small (20 * 20 to 200 * 200) Hermitian matrices. These ...
Scriabin's user avatar
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Relationship between two matrices whose sum and difference are similar matrices

Let $A$ and $B$ be hermitian $n$-by-$n$ matrices with complex coefficients with the property that $A+B$ and $A-B$ are similar, i.e. there is a unitary $U$ such that $A+B = U(A-B)U^\dagger$. Does it ...
extempore's user avatar
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Finding Correspondence between Matrices after Decomposing a Hermitian Matrix

After decomposing the Hermitian matrix $M$, I obtain a set of matrices $D_j$, where each $D_j$ is defined as $D_j = \lambda_j u_j u_j^H$, $\lambda_j$ and $u_j$ are eigenvalues and eigenvectors, and $M$...
Cyberturist's user avatar
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2 answers
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Eigenvalues and eigenvectors of certain rank two Hermitian matrix

Let $n\geq 2$ and $A\in\mathbb{C}^{n\times n}$ such that $$ A = \begin{bmatrix} 0 & x^*\\ x & 0 \end{bmatrix} $$ for some $x\in\mathbb{C}^{n-1}\backslash\{0\}$. Is there an explicit form for ...
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Show that operator is self-adjoint

Show that the operator B in a space of $2 \times 2$ real matrices is self-adjoint $$ BX = X \begin{pmatrix} 1 & 2 \\ 2 & 4 \\ \end{pmatrix} $$ I attempted to apply the theorem that an ...
Gleb Cloudy's user avatar
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Conditions for a Hermitian matrix to be positive definite

It's a question from chapter 7c of Linear algebra done right 4th edition. Suppose $T \in \mathcal{L}(V)$ and $e_1 \cdots e_n$ is an orthonormal basis of $V$. Prove that T is a positive operator if ...
Azuuu's user avatar
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Inequalities on the trace of matrix products

For two $n\times n$ hermitian matrices $A$, $B$, we have the trace inequality $$\text{tr}(AB)\leq\sum_{i=1}^{n}\lambda_i(A)\lambda_i(B)$$ where the $\lambda_i(X)$ are the eigenvalues of X ordered in ...
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Generation of Hermitian invertible matrix with fixed number of non-zero elements

I have a Hermitian matrix that is invertible, that is I can write it as: $H = U^\dagger D U$, where $U$ is a unitary matrix, and $D$ is a diagonal matrix with the eigenvalues of $H$, which must be ...
Damuna Taliffato's user avatar
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If $\rho_{AB}$ is a separable then the partial transpose w.r.t to A is PSD

Def: The partial transpose of a linear operator $\rho_{AB}$ over a Hilbert space $H_A \otimes H_B$ w.r.t A is defined for a linear operator $\rho_{AB}=\rho_A \otimes\rho_B$ as $\rho^{T_A}_{AB}=\rho_A^...
some_math_guy's user avatar
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Is the Eigen spectrum of a matrix completely defined by the algebra of its parts?

Consider two vector spaces $\mathbb{C}^n$ and $\mathbb{C}^m$, where $0<n<m<\infty$. Now, I'd like to define matrices $A\in\mathbb{C}^{n\times n}$ and $B\in\mathbb{C}^{m\times m}$ in the ...
Jun_Gitef17's user avatar
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3 answers
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The relation between the eigenvalue of a Hermitian matrix and the block matrix that composed by it real and imaginary part

Recently I am reading a paper. In their "Proof of Lemma 1" on page 24, they have: $$\lambda_+(\mathbf{Q})=2\lambda_+(\tilde{\mathbf{Q}})$$ where $\mathbf{Q}$ is a Hermtian matrix, $\tilde{\...
tyrela's user avatar
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Trace of product of three matrices

I have an expression of the form \begin{equation} i\operatorname{tr}(ABC - B^\dagger AC), \end{equation} where $A$ and $C$ are Hermitian matrices, but $B$ is not. $B-B^\dagger := i D$, where $D$ is ...
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Decomposition of a positive semidefinite matrix in the form $Q = BB^H = CC^H$

Let $Q \in \mathbb{C}^{m \times m}$ be a given positive (semi)-definite matrix such that $Q = CC^H,$ for some known matrix $C \in \mathbb{C}^{m \times r}$ having full column rank and orthogonal ...
ketan bapat's user avatar
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How to prove $SUS=\frac{(S|S)}{2}U$ with $S$ and $U$ are skew-hermitian with following conditions?

We denote the $(A|B):=-{\rm tr}(AB)$ as the inner product of the skew-hermitian matrix, which is easy to prove by the definition. I'm stuck by the problem when I read the Lie algebra of matrix group, ...
Krystal Justin's user avatar
2 votes
1 answer
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Step in derivation of Hadamard's First Variational Formula for Hermitian Matrices

This arose in trying to understand the details of a proof of Hadamard's first variational formula in Terence Tao's "Topics in Random Matrix Theory". Suppose $A$ is an $n \times n$ Hermitian ...
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Why does $\langle x\vert A^\dagger A\vert x\rangle$ being independent of $x$ imply this Hermitian property?

This property seems to be used in the proof of Theorem 3 of these notes. In particular, see (3.4) and (3.5). Suppose that we have a complex finite dimensional Hilbert space $\mathcal{H}$ and consider ...
user1936752's user avatar
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Can the product of a complex symmetric unitary matrix and a skew-hermitian matrix be complex skew-symmetric?

Let $S$ be a nonzero complex symmetric unitary matrix ($S\neq 0$, $S = S^T$, $S^HS=Id$) Let $B$ be a nonzero skew-hermitian matrix ($B\neq 0$, $B^H = -B$) Can their product $P\triangleq SB$ be ...
LucLeMag's user avatar
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Conjugate of an operator and its relation with its squaring value

This question is straightforward. Is it possible for a unit length operator (vector, multivector, etc.), excepts for $-1$ itself that squares to $+1$ to have its hermitian conjugate equal to $-1$?, i....
physicsrev's user avatar
1 vote
1 answer
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Condition on unitary operator for real eigenstates of Hamiltonian

I'm working with the discrete-time quantum walk in which the evolution is described by the unitary operator - $$U = S(I\otimes C)$$ where $C$ is the coin operator (acts on spin degree of freedom of ...
Young Kindaichi's user avatar
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Inverse of a special hermitian sparse matrix

While exploring some data modelling, I observed without being able to prove it that the inverse of matrix $A$ of the following form: it is hermitian, i.e. $A_{ij}=A^{*}_{ji}$ it has non-zero positive ...
Riccardo Buscicchio's user avatar
2 votes
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Set of positive definite Hermitian matrices as quotient

Let $$\mathcal{H}_n^{++} := \{ A \in \mathrm{Mat}_n(\mathbb{C}) \vert A = A^{\dagger} \text{ and } A > 0 \}$$ denote the space of positive definite Hermitian matrices. I have read in some ...
KuSi's user avatar
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Do all symmetric matrices represent a quadratic form?

If a quadratic form is represented as a matrix, it may not be symmetric, but there must exist a representation that is symmetric (or Hermitian if complex). But is the reverse true? That is, do all ...
Davey's user avatar
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1 answer
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Eigenvalues of matrix with submatrices

So, I was doing homework and I had to turn an arbitrary invertible $N \times N$ matrix $A$ into a Hermitian $2N \times 2N$ matrix $A'$. This is rather simple, namely pick $A' = \begin{pmatrix} O &...
Tipeg's user avatar
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generalized matrix inequality for complex Hermitian matrices

Assume having a symmetric real matrix $A$ and a skew-symmetric matrix $\Delta = [0 1; -1, 0 ]$, such that the following generalized matrix inequality holds in the PSD sense: $$\pm \frac{i}{2} \Delta\...
hafezmg48's user avatar
2 votes
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Closedness of a cone

Let $N\in\mathbb{N}$ and $P$ be a symmetric pattern; i.e. $P$ is a subset of $\{1,\dots,N\}\times\{1,\dots,N\}$ such that $(i,i)\in P$, for all $i\in\{1,\dots,N\}$, and $(i,j)\in P$ if and only if $(j,...
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Difference Between Simultaneous Diagonalization and Generalized Eigenvalue Problem

I am wondering about the difference between simultaneous diagonalization and the general eigenvalue problem. Here is my understanding of simultaneous diagonalization and the general eigenvalue problem:...
user19402204's user avatar
3 votes
1 answer
179 views

Eigenvalue relation of a symmetric matrix $A$ and $A + vv^T$

My question pertains to the material in the book "The Algebraic Eigenvalue Problem" by J.H. Wilkinson. Section "Symmetric matrix of rank unity", pages 96-97. The setup is as ...
MonteNero's user avatar
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Cauchy-Schwarz-like inequality for certain determinants

Let $U = [u_1 | \ldots | u_n]$ be a special unitary matrix (meaning that $u_i$ is its $i$-th column) and let $A$ be an $n$-by-$n$ hermitian matrix, where $n \geq 3$. I am interested in the following &...
meler's user avatar
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Inequality involving minors of a hermitian matrix

Let $A$ be an $n \times n$ hermitian matrix with $n \geq 3$. I am trying to prove the following inequality involving its minors $$\left| \sum_{k=3}^n A_{3k} A_{[12k],[123]} \right| \leq \sqrt{\sum_{i &...
meler's user avatar
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If $A$ is diagonalizable and $B$ commutes with all transformations commuting with $A$, does it follow that $B=p(A)$? [duplicate]

Let $A$ be a diagonalizable transformation on a finite-dimensional vector space. If $B$ commutes with all transformations commuting with $A$, does it follow that $B=p(A)$ for some polynomial $p$? ...
Milten's user avatar
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Why the real space of Hermitian matrices is composed of space of traceless Hermitian matrices direct sum $\text{Span}_{\mathbb{R}}(\{\mathbb{I}_X\})$

I'm trying to understand a so-called fact (From the SM of article, proof of Lemma-SM 3) that $$\mathcal{h}(X)=\mathcal{h}_0(X)\oplus \text{Span}_{\mathbb{R}}(\{\mathbb{I}_X\}),$$ where $h(X)$ means ...
hui's user avatar
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Need a book about complex hermitian matrix and special unitary matrices and their optimization in numerical problems

I am studying about tomographic interferometric sar polarimetry and i don't have enough knowledge about complex hermitian matrix , special unitary groups, their quadratic forms and numerical ...
mohammad sadeg 's user avatar
2 votes
1 answer
68 views

Show that $B\ge A$ and $B\ge -A$ implies $B\ge |A|$ for self-dual matrices.

EDIT: It turns out, somewhat embarrassingly, that I misread the source exercise. The true claim is that $|A|$ is the smallest Hermitian transformation that commutes with $A$ and satisfies $|A|\ge\pm ...
Milten's user avatar
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Matrix multiplication involving a unitary matrix in exponential form and a simple matrix

I am not sure if this is even a valid question, but it will help to clear my doubts anyway. For context, check out Violating the Thermodynamic Uncertainty Relation in the Three-Level Maser. So I am ...
QuestionTheAnswer's user avatar
1 vote
1 answer
162 views

Unsymmetric to Symmetric Tridiagonal Matrix

According to the Wikipedia page here, a real, unsymmetric tridiagonal matrix can be brought to symmetric form by a similarity transform. Does anyone know if a generalization of the formula given there ...
CW279's user avatar
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How to approximately diagonalize a special symmetric hermitian matrix?

Given a hermitian matrix $H$ as follows: \begin{equation} H = \begin{bmatrix} H^1 & V^{12} \\ V^{21} & H^2 \end{bmatrix}. \end{equation} Here, $H^1,H^2\in\mathbb{C}^{N\times N}$ ...
bb wang's user avatar
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Bounding the Norm of a Nested Commutator

Consider a Hermitian matrix $B$ and a set of Hermitian matrices $\{A_i\}_{i=1}^n$. Now consider the nested commutator: $$ [A_n, [A_{n-1}, \dots [A_1,B]]. $$ I want to put an upper bound on the ...
Hans Schmuber's user avatar
1 vote
2 answers
208 views

Help me to improve this proof of the spectral theorem of Hermitian matrices

Spectral theorem for Hermitian matrices states that, for an $n\times n$ Hermitian matrix $A$: a) all eigenvalues are real, b) eigenvectors corresponding to distinct eigenvalues are orthogonal, c) ...
Aris Makrides's user avatar
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0 answers
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Proof that an operator is self-adjoint if and only if its matrix is self-adjoint

As stated in the wikipedia page on self-adjoint operators, $A$ is a self-adjoint operator on an finite-dimensional inner product space $V$ if and only if, given an orthonormal basis, the matrix of $A$ ...
Sam's user avatar
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1 answer
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Prove: If matrix $A\in\mathbb{C}^{m\times n}$ with $m>n$ is full rank and $B\in\mathbb{C}^{n\times n}$ is diagonal, $ABA^H$ is positive semi-definite

If matrix $A\in\mathbb{C}^{m\times n}$ with $m>n$ is full rank and $B\in\mathbb{C}^{n\times n}$ is diagonal with $det(B)\ne0$, and $(B)_{ii}>0$ how can we prove that $ABA^H$ is positive semi-...
Squid49134's user avatar
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3 answers
82 views

Prove: For full rank matrix $A\in\mathbb{C}^{m\times n}$ with $m>n$ and diagonal matrix $B\in\mathbb{C}^{n\times n}$ we have $rank(ABA^H)=n$

For full rank matrix $A\in\mathbb{C}^{m\times n}$ with $m>n$ and diagonal matrix $B\in\mathbb{C}^{n\times n}$ with $det(B)\ne0$ how can we prove that $rank(ABA^H)=n$? I have already proven that $...
Squid49134's user avatar
3 votes
1 answer
160 views

Eigenvalue problem and its complex conjugate

The following problem comes from a classical-mechanics treatment of lattice vibrations. It's essentially a linear algebra question, though, so I thought it would be on-topic here. If it may be assumed ...
CW279's user avatar
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1 vote
1 answer
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Let $A$ be a Hermitian matrix, prove that $v^*Av >0$ for all $0 \not = v \in \Bbb C^n$ $\iff$ all the eigenvalues of $A$ are greater than zero

Let $A \in \Bbb C^{n \times n}$ be a Hermitian matrix, prove that $v^*Av >0$ for all $0 \not = v \in \Bbb C^n$ $\iff$ all the eigenvalues of $A$ are greater than zero I started this by stating ...
Adamrk's user avatar
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1 vote
2 answers
58 views

ordering of eigenvalues for simultaneously diagonalizable matrices [closed]

Suppose I have two commuting Hermitian matrices $A$ and $B$: $[A,B] = 0$. I can always find a unitary operator $U$ such that simultaneously diagonalize both matrices, i.e., \begin{equation} U^* A U = ...
Hailey Han's user avatar
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1 answer
180 views

If $A$ is Hermitian and $U=(A-iI)(A+iI)^{−1}$ is unitary, then $U-I$ is invertible

I'm stuck on the $3$rd part of a question, the first part was proving $A-iI$ is invertible when $A$ is Hermitian, second was proving $U=(A-iI)(A+iI)^{−1}$ is unitary. Now I'm being asked to prove $U-I$...
shai's user avatar
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