I have a question regarding self-adjoint(hermitian) matrices, and their properties when multiplied by an imaginary number. The matrix in question is: $$\mathbf{A} = c \begin{bmatrix} 0&0&0&0\\ 0&0&2i&0\\ 0&-2i&0&0\\ 0&0&0&0 \end{bmatrix}\\ \textit{where} \; \mathbf{A}^\dagger=\mathbf{A} $$ This should be pretty simple to answer to, but I am having a brain fart at the moment: if we assume $c=\frac{i}{2}$, then: $$\mathbf{A} = \begin{bmatrix} 0&0&0&0\\ 0&0&-1&0\\ 0&1&0&0\\ 0&0&0&0 \end{bmatrix}\\ \textit{where} \; \mathbf{A}^\dagger\neq\mathbf{A} $$ Do hermitian operators stop to be so if multiplied by a complex number?
2026-03-26 17:46:18.1774547178
Hermitian Matrix multiplied by complex numbers
854 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in MATRICES
- How to prove the following equality with matrix norm?
- I don't understand this $\left(\left[T\right]^B_C\right)^{-1}=\left[T^{-1}\right]^C_B$
- Powers of a simple matrix and Catalan numbers
- Gradient of Cost Function To Find Matrix Factorization
- Particular commutator matrix is strictly lower triangular, or at least annihilates last base vector
- Inverse of a triangular-by-block $3 \times 3$ matrix
- Form square matrix out of a non square matrix to calculate determinant
- Extending a linear action to monomials of higher degree
- Eiegenspectrum on subtracting a diagonal matrix
- For a $G$ a finite subgroup of $\mathbb{GL}_2(\mathbb{R})$ of rank $3$, show that $f^2 = \textrm{Id}$ for all $f \in G$
Related Questions in COMPLEX-NUMBERS
- Value of an expression involving summation of a series of complex number
- Minimum value of a complex expression involving cube root of a unity
- orientation of circle in complex plane
- Locus corresponding to sum of two arguments in Argand diagram?
- Logarithmic function for complex numbers
- To find the Modulus of a complex number
- relation between arguments of two complex numbers
- Equality of two complex numbers with respect to argument
- Trouble computing $\int_0^\pi e^{ix} dx$
- Roots of a complex equation
Related Questions in SELF-ADJOINT-OPERATORS
- Why the operator $T$ is positive and self-adjoint, which $(T(t)f)=\sum_{n=0}^{\infty}(n+1)^{-t}c_{n}z^n$?
- Express in terms of $E$ a self-adjoint operator $T$ such that $T^2 = I+E$
- Showing $(1-x^2)u''-xu'+9u=x^3$ is formally self-adjoint
- Adjoint relation: transpose or conjugate transpose?
- Dimension of the null space of a compact perturbation of a self-adjoint operator
- Proof of a linear algebra lemma for Cohn-Vossen's theorem
- Fredholm Alternative for Singular ODE
- Let A be a self-adjoint, compact operator on a Hilbert space. Prove that there are positive operators P and N such that A = P − N and P N = 0.
- Convergence of (unbounded) self-adjoint operators
- Eigendecomposition of Self-Adjoint Operator with Non-Positive Inner Product
Related Questions in HERMITIAN-MATRICES
- Let $M$ be an $n \times n$ complex matrix. Prove that there exist Hermitian matrices $A$ and $B$ such that $M = A + iB$
- Product of two Hermitian matrices
- Eigenvectors of a Hermitian matrix
- Does matrix modulus satisfy triangle inequality for Loewner order?
- Prove sum of products of Hermitian matrices to be Hermitian
- What is dimension over $\mathbb R$ of the space of $n\times n$ Hermitian matrices?
- Nearest positive semidefinite matrix to a complex-valued Hermitian matrix
- SVD for Hermitian Matrix
- How does one prove the determinant inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?
- A normal matrix with real eigenvalues is Hermitian
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
When $A$ is Hermitian, we have $(cA)^\dagger-cA=c^\dagger A^\dagger-cA=(c^\dagger-c)A$. Therefore $cA$ is Hermitian if and only if $A=0$ or $c$ is real.