Questions tagged [hermitian-matrices]

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

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Voisin 3.1.1 Hermitian Geometry

I am a bit confused about what appears to be basic linear algebra in Voisin's book. I suspect I am misunderstanding maybe some of the notations. My first confusion is from this passage. $W_{\mathbb{C}...
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What conditions are required to guarantee that my matrix is skew-Hermitian?

Consider the equation $$ \frac{\partial \boldsymbol{T}}{\partial t}=\kappa \space \frac{\partial^2 \boldsymbol{T}}{\partial x^2} $$ with $$ \boldsymbol{T}=(T_1,T_2,...T_N)^T $$ Let $$ T_i=\sum_{j=1}^...
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Determining the signature of a Matrix based on the characteristic polynomial.

Let $A$ be a hermitian matrix with the characteristic polynomial $p_A=a_0+a_1x+...+a_nx^n.$ Furthermore let $p$ be the number of sign changes in the sequence $\{a_0,a_1,...a_{n-1},1\}$ and $q$ be the ...
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Calculating the signature of two hermitian forms [closed]

I have a simple question regarding the calculation of the signature of two hermitian forms. $L:\mathbb{C^4}\times\mathbb{C^4}\rightarrow \mathbb{C}$ with $L\bigg( \begin {pmatrix} x_1\\y_1\\z_1\\t_1\...
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Hermitian matrix & Complex Numbers

Please help with the following two questions. Many thanks.
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Diagonal element of traceless hermitian matrix?

In physics, we are familiar with a set of traceless hermitian matrices named Pauli matrices: $$ {\displaystyle {\begin{aligned}\sigma _{1}=\sigma _{\mathrm {x} }&={\begin{pmatrix}0&1\\1&0\...
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How to extract $\mathbf b$ from $M = a I + \mathbf b \cdot \mathbf S$?

Any $2 \times 2$ hermitian matrix $M$ can be written $$M = a I + \mathbf b \cdot \boldsymbol \sigma,\tag{1}$$ where $a \in \mathbb R$, $\mathbf b \in \mathbb R^3$, $\boldsymbol \sigma$ is the Pauli ...
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Probability density function invariant under unitary transformation

Recently I have been studying Mehta's book on Random Matrices (3rd edition). In this book the author defines the Gaussian Unitary Ensemble in the set of hermitian matrices with 2 specific properties. ...
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Hilbert basis is this definition correct? Can't we get any simpler

There is a definition of Hilbert basis in this paper (chapter 1.2) that says : $(e_{i})$ is a Hilbert basis iff $$ (1). (e_{i}\dagger e_{j})=\delta_{i,j}$$ $$(2). \Sigma e_{i}e_{i}\dagger = I$$ where $...
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Inequality for Hermitian matrices

Does the following equality hold for $A,B$ ( both Hermitian matrices): $$\vert \text{tr}(AB) \vert \leq \displaystyle\left(\text{tr}(\vert A\vert^k) \text{tr}(\vert B \vert ^k)\right)^{\frac{1}{k}}?$$ ...
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A question on inequalities associated to hermitian positive semidefinite matrices

Let $\mathcal{H}_N$ denote the cone of hermitian $N$ by $N$ positive semidefinite matrices and let $H \in \mathcal{H}_N$. We associate to $H$ the following complex numbers: \begin{align*} c_1 &= ...
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Hermitian matrix has a symmetric spectrum if and only if $\lambda_1=\lambda_n$?

I seem to remember a result claiming that an (irreducible) zero-diagonal $n\times n$ Hermitian matrix $H$ with entries $h_{ij}\in \{z\in\mathbb{C}~:~ |z|=1\}\cup\{0\}$ has eigenvalues $\lambda_1\geq\...
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Is this rank-$1$ (complex) matrix positive semidefinite?

In Is this rank-$1$ matrix semidefinite?, I have seen that $X = xx^T$ is PSD when $x$ is real. What about the case when $X$ is Hermitian? I know that it is PSD but I'm not exactly sure how to prove it....
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Find a matrix of a hermitian form

Let's $\phi : \mathbb{C}\times\mathbb{C} \to \mathbb{C}$ defined by $$\langle x, y \rangle = \sum_{k=1}^3 \overline{x_k}y_k + \frac{1}{2} \sum_{1\leq j < k \leq 3} \overline{x_j}y_k+\overline{x_k}...
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How to prove that changing the equality constraints does not affect the sign of the optimal value of the objective function?

Given random Hermitian matrices $A_1,A_2,A_3,A_4$,they satisfy: $$ \text{tr}(A_i)\in[0,1],\quad i=1,2,3,4.\quad \text{tr}(A_1+A_2)=\text{tr}(A_3+A_4)=1 $$ Given Hermitian variables $X_1,X_2,X_3,X_4$, ...
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Are anticommuting involutory matrices Hermitian?

If $\alpha$ and $\beta$ are matrices such that $\alpha^2=\beta^2=I$, and they anticommute, i.e $\{\alpha,\beta\}=0$. Then, does this imply that $\alpha$ and $\beta $ are Hermitian?
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Hermitian matrix having repeated eigenvalues

Let $A$ be a Hermitian matrix ($A^{*}=A$) then, by the spectral theorem there exists an orthonormal basis $\left(e_{i}\right)_{1\leq i\leq n}$ of $\mathbb{C}^{n}$ consisting of eigenvectors of $A$. ...
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Semipositive definite and hermiticity implication

Its something that is in all basic algebra books that, a semipositive definite operator is Hermitian and the eigenvalues of this hermitian operator are positive. But I couldn't find any place where ...
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Low-rank approximation of Hermitian matrix with given spectral decomposition

Given a Hermitian matrix $C^{n \times n}$ with a known spectral decomposition $U \Delta U^{-1}$. Is there any way to do a low-rank approximation of $H$ without computing the SVD of $H$ from scratch?
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Hermitian and Observables

Let $H$ be an finite dim Hilbert space with $|\phi \rangle$ in H, $\langle\phi \mid \phi\rangle=1$, and $\rho \triangleq|\phi\rangle\langle\phi| .$ Let $A$ be an observable which is represented by the ...
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Range of values for $\frac{\langle f | \mathbf n \cdot \boldsymbol \sigma |f \rangle}{\langle f | f \rangle}$, with $|\mathbf n| = 1$?

I'm trying to find the range of possible values for the expression $$w := \frac{\langle f | \mathbf n \cdot \boldsymbol \sigma |f \rangle}{\langle f | f \rangle},$$ where $\mathbf n \in \mathbb R^3$, $...
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On permuting the eigenvectors of a regular hermitian matrix while keeping the eigenvalues fixed

Let $H$ be a "regular" $n$ by $n$ hermitian matrix, where by "regular", we mean that it has distinct eigenvalues. We can then write $H$ as $$ H = UDU^*, $$ where $U$ is unitary and ...
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How to show that product of hermitian matrixes have common eigenvector

I need to show that two hermitian matrices A and B have a common ON-basis of eigenvectors (i.e. they can be diagonalized, A=UD1U* and B=UD2U* with the same unitary U) if and only if the commute (i.e. ...
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Geometry of Skew-Hermitian matrices

In the question I will be referring to finite dimensional complex vector spaces. I know that the eigenvalues of an Skew-Hermitian matrix $S$ are purely imaginary, but what is their geometric effect on ...
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Matrices that are simultaneously Hermitian and unitary [duplicate]

My quantum mechanics professor was discussing the properties of Pauli matrices, their being both Hermitian and unitary. Then he made a remark that it is not possible to find three $n \times n$ ...
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Unitary equivalence of sums of unitary equivalent hermitian matrices

Consider two hermitian matrices $A$ and $B$. Suppose that there exists a unitary matrix $U$ such that $A+B$ is unitarily equivalent to $U A U^* +B$. Does this imply that there exists a unitary matrix $...
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Updating degenerate Eigenvectors

I have a Hermitian Matrix $H$ that evolves very slightly over ~50 iterations until convergence of a Schrödinger/Poisson system is achieved. The eigenvalues I'm interested in are $\lambda_1$ and the ...
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How to extract the vector from a rank-$1$ matrix? [closed]

Given Hermitian and positive semidefinite rank-$1$ matrix ${\bf Z} \in \mathbb C^{N \times N}$, how to find vector $\mathbf z \in \mathbb C^{N \times 1}$ such that $\mathbf Z = \mathbf z \mathbf z^{H}$...
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Cholesky decomposition for a Hermitian matrix in SDP

I have a variable matrix $W$ that is Hermitian and is used in two SDP problems. Problem 1 has constraints that depend on the real diagonal elements of $W$. Example of the constraint is $W_{ii}+x_{ij}...
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Geometric interpretation of adjoint

Given the inner product $\langle x, y \rangle = y^HMx, \, x,y\in\mathbb{C}^n$ where $y^H = \bar{y}^T$ is the conjugate transpose and $M^H=M$ positive definite, I find the adjoint $A^*$ of $A\in\mathbb{...
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Diagonalization of Hermitian Matrices

According to Principles of Quantum Mechanics , Shankar Reference page 40 we can diagonalize the orthonormal basis $|i \rangle$ in eigenbasis $|\omega_i \rangle$ by the simple $|\omega_i \rangle= U |i \...
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Making complex matrix real with unitary transformation

Consider the Hermitian matrices: $$A=\begin{bmatrix} 0 & -0.7011-0.0912i & 0 \\ -0.7011+0.0912i & 0 & 0.6702-0.2255i\\ 0 & 0.6702+0.2255i & 0 \end{bmatrix}\\ B=...
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A is a Hermitian matrix ,Is a matrix U diagonizable if U(A-iI)=(A+iI) [closed]

We know that $A$ is a Hermitian matrix .So $A\pm iI$ are invertible. How do I prove that $U$ is diagonal given $U(A-iI)=(A+iI).$ I tried claiming that $A\pm iI$ is Hermiatian but it is not not. I ...
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Checking the criterion for the positive definiteness of a matrix

The article Bounds for Elgenvalues Using Traces on page 482 has the following statement: If $A$ is Hermitian, the inequality (2.36) shows that $A$ is positive definite when $$\frac{\mbox{Tr}(...
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prove the matrix is negative semi-definite

Given $a_1+a_2+\cdots+a_n=0$ and $a_i^2+b_i^2=1,\forall i\in\{1,2,\cdots,n\}$, prove the following matrix is negative semi-definite $$\left[\begin{array}{ccccc|c} -b_1&0&\cdots&0&0&...
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Show that $\langle {\bf{x}^{\prime}} |\hat P^{\dagger}\hat P|{\bf{x}}\rangle=\langle\hat P{\bf{x}^{\prime}}|\hat P{\bf{x}}\rangle$ using matrices

From my lecture notes, I have that Where the missing equations are $$\hat P {\bf{x}}=-{\bf{x}}\tag{6.23}$$ and $${\hat P}^2 {\bf{x}}=\hat P\left(\hat P {\bf{x}}\right)=\hat P\left(-{\bf{x}}\right)={\...
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Integrals Involving Exponential and Determinant

Suppose $a_{ij}=a_{ij}(x_1, x_2, \ldots, x_n)$ are symmetric functions in the variables $x_1, x_2, \ldots, x_n$. I'm wondering if there is a general method or approach for evaluating integrals of the ...
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Number of free real parameters in Hermitian vs unitary matrix

There seems consensus that the number of free real parameters of an $n \times n$ Hermitian matrix $M$ is $n^2$; see e.g. this post and also this short note. On the other hand, such a matrix has a ...
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Is the Fourier transform of real input truly Hermitian?

I asked a question on stackoverflow and was advised to repost it here with some additional context (it was originally a coding question about NumPy FFTs, but was suggested the underlying question is ...
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Spectrum of tensor product symmetric hermitian map ${\rm A}=\sum_l\mathbf e_l\mathbf e_l^*\otimes{\rm B}^l+{\rm B}^l\otimes\mathbf e_l\mathbf e_l^*$

Consider the hermitian linear map $\mathrm A:V\otimes V\to V\otimes V$ (where $V$ is a finite-dimensional vector space) given by the matrix $$ \mathrm A = \sum_l \mathbf e_l\mathbf e_l^*\otimes\mathrm ...
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Why do we define Symplectic groups with Transpose and not hermitian adjoint?

In the definition of a symplectic group over a field, we take the definition: all matrices $S$ of some dimension $2n$, such that $$S^T \Omega S = \Omega$$ where omega is a skew-symmetric matrix (bi-...
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Proof that the determinant of the Hermitian conjugate of $A$ is the complex conjugate of the determinant of $A$

While revising for an exam, I came across the following proof that the determinant of a unitary matrix $A$ is $\pm 1$: $$1=\det(I)=\det(A^{\dagger}A) = \det(A)^{*}\det(A)=|\det(A)|^2 $$ This seems to ...
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Eigenvalues/vectors of combinations of Gell-Mann matrices

Let's consider the Gell-Mann matrices $\vec{\lambda} =(\lambda_1, \lambda,_2,\cdots, \lambda_8)$. Another hermitian matrix can be constructed in terms of a real vector $\vec{a}=(a_1,\cdots,a_8)$ by $A ...
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Reconstruct a matrix by its Hermitian part

Suppose we have a matrix $A \in M_{n\times n}(\mathbb C)$. It is well-known that we can write $A$ as $$ A=H(A)+iK(A) $$ where $H(A)=\frac 1 2 (A+\bar A^t)$ is an Hermitian matrix and $K(A)= \frac 1 2 (...
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Hermitian Matrix eigenvalues

I am trying to show the following sentence: Let H be hermitian with $\sigma(H) \subseteq \lbrace-r,r\rbrace$. Show that H ∘ H = $r^2$* I. Since H is hermitian, we know that we can decompose the matrix ...
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Corollary of Schur-Horn

Let $M \in \mathbb{C}^{n \times n}$ be a Hermitian matrix, then $$\sum_{i=1}^k \tilde{m}_{ii} \leq \sum_{i=1}^k \lambda_i,$$ for $k = 1,\dots,n-1$. Here $\tilde{m}_{11} \geq \tilde{m}_{22} \geq \cdots ...
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Can we construct an ORTHOGONAL ($trace(A^\dagger B) = 0$) basis for Hermitian matrices made of PSD (positive semi-definite) Hermitian matrices?

The space of $N \times N$ Hermitian matrices can be construed as a real vector space with dimension $N^2$. One can define an inner product on this real vector space by $\langle A, B \rangle := trace(A^...
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Minimal spanning set ("conical basis") for 2x2 Hermitian PSD (positive semi-definite) cone?

A linear combination $ax + by$ is called conic(al) if $a, b \ge 0$ (cf. section 2.1.5 of Boyd, Vandenberghe). I.e. conic(al) combinations are just linear combinations where the coefficients are ...
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If two hermitian matricies commute, do they share an orthonormal eigenbasis?

Heads up: I'm a physicist. I know that if $A=A^\dagger$, $B=B^\dagger$ commute ($AB=BA$) then there is an eigenbasis $\{v_1,v_2,...,v_n\}$ which diagonalises both $A$ and $B$ even if they are ...
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Change of basis matrix exactly one row away from being correct

For homework, I'm given a matrix $$ A = \begin{bmatrix} 3 & 2i & -2i\\ -2i & 0 & -1\\ 2i & -1 & 0 \end{bmatrix} $$ in an hermitian space. We are trying to find an orthonormal ...
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