If S is any $n \times n$ real, symmetric, invertible matrix and D is any $n \times n$ diagonal matrix such that $0\prec D \prec I$ then does there exist a constant $\gamma$ such that:
$trace(S^{-1}DS^2DS^{-1})\leq \gamma \cdot n$
2026-03-31 03:35:43.1774928143
trace inequalities: linear algebra
163 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in LINEAR-ALGEBRA
- An underdetermined system derived for rotated coordinate system
- How to prove the following equality with matrix norm?
- Alternate basis for a subspace of $\mathcal P_3(\mathbb R)$?
- Why the derivative of $T(\gamma(s))$ is $T$ if this composition is not a linear transformation?
- Why is necessary ask $F$ to be infinite in order to obtain: $ f(v)=0$ for all $ f\in V^* \implies v=0 $
- I don't understand this $\left(\left[T\right]^B_C\right)^{-1}=\left[T^{-1}\right]^C_B$
- Summation in subsets
- $C=AB-BA$. If $CA=AC$, then $C$ is not invertible.
- Basis of span in $R^4$
- Prove if A is regular skew symmetric, I+A is regular (with obstacles)
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 INEQUALITY
- Confirmation of Proof: $\forall n \in \mathbb{N}, \ \pi (n) \geqslant \frac{\log n}{2\log 2}$
- Prove or disprove the following inequality
- Proving that: $||x|^{s/2}-|y|^{s/2}|\le 2|x-y|^{s/2}$
- Show that $x\longmapsto \int_{\mathbb R^n}\frac{f(y)}{|x-y|^{n-\alpha }}dy$ is integrable.
- Solution to a hard inequality
- Is every finite descending sequence in [0,1] in convex hull of certain points?
- Bound for difference between arithmetic and geometric mean
- multiplying the integrands in an inequality of integrals with same limits
- How to prove that $\pi^{e^{\pi^e}}<e^{\pi^{e^{\pi}}}$
- Proving a small inequality
Related Questions in TRACE
- How to show that extension of linear connection commutes with contraction.
- Basis-free proof of the fact that traceless linear maps are sums of commutators
- $\mathrm{tr}(AB)=\mathrm{tr}(BA)$ proof
- Similar 2x2 matrices of trace zero
- Basis of Image and kernel of Linear Transformation $\mathbb(M_{2,2})\rightarrow\mathbb(R^3) = (trace(A), 5*Trace(A), - Trace(A))$
- Replace $X$ with $\mbox{diag}(x)$ in trace matrix derivative identity
- Proving that a composition of bounded operator and trace class operator is trace class
- If $A \in \mathcal M_n(\mathbb C)$ is of finite order then $\vert \operatorname{tr}(A) \vert \le n$
- Characterisations of traces on $F(H)$
- "Symmetry of trace" passage in the proof of Chern Weil.
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?
If you are asking whether there exists a constant $\gamma$, that is independent of $n,S$ and $D$, such that $\operatorname{trace}(S^{-1}DS^2DS^{-1})\le\gamma\,n$, the answer is no. Let $x>0$ and $$ D=\frac14\pmatrix{1\\ &2},\ U=\frac1{\sqrt{2}}\pmatrix{1&1\\ 1&-1},\ S=U\pmatrix{1\\ &x}U^T. $$ Then $S^{-1}DS=\frac14U\pmatrix{\frac32&-\frac x2\\ -\frac1{2x}&\frac32}U^T$ and hence $$ \operatorname{trace}(S^{-1}DS^2DS^{-1}) =\left\|\frac14\left(\frac32,\,-\frac1{2x},\,-\frac x2,\,\frac32\right)\right\|^2, $$ which is unbounded as $x\to0$ or $x\to\infty$.