I'm trying to show the following: $$\frac{1}{2}(x^{T}Ax +y^{T}Ay) \leq x^{T}Ay$$ with A symmetric and $||x||_{2}, ||y||_{2} \leq 1$. I'm not sure if it's true or not and was wondering if anyone had any insight into this? Thanks a ton!
2026-04-04 12:06:50.1775304410
Average of Two Quadratic Forms
44 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 CONVEX-OPTIMIZATION
- Optimization - If the sum of objective functions are similar, will sum of argmax's be similar
- Least Absolute Deviation (LAD) Line Fitting / Regression
- Check if $\phi$ is convex
- Transform LMI problem into different SDP form
- Can a linear matrix inequality constraint transform to second-order cone constraint(s)?
- Optimality conditions - necessary vs sufficient
- Minimization of a convex quadratic form
- Prove that the objective function of K-means is non convex
- How to solve a linear program without any given data?
- Distance between a point $x \in \mathbb R^2$ and $x_1^2+x_2^2 \le 4$
Related Questions in QUADRATIC-FORMS
- Can we find $n$ Pythagorean triples with a common leg for any $n$?
- Questions on positivity of quadratic form with orthogonal constraints
- How does positive (semi)definiteness help with showing convexity of quadratic forms?
- Equivalence of integral primitive indefinite binary quadratic forms
- Signs of eigenvalues of $3$ by $3$ matrix
- Homogeneous quadratic in $n$ variables has nonzero singular point iff associated symmetric matrix has zero determinant.
- Trace form and totally real number fields
- Let $f(x) = x^\top Q \, x$, where $Q \in \mathbb R^{n×n}$ is NOT symmetric. Show that the Hessian is $H_f (x) = Q + Q^\top$
- Graph of curve defined by $3x^2+3y^2-2xy-2=0$
- Question on quadratic forms of dimension 3
Related Questions in SYMMETRIC-MATRICES
- $A^2$ is a positive definite matrix.
- Showing that the Jacobi method doesn't converge with $A=\begin{bmatrix}2 & \pm2\sqrt2 & 0 \\ \pm2\sqrt2&8&\pm2\sqrt2 \\ 0&\pm2\sqrt2&2 \end{bmatrix}$
- Is $A-B$ never normal?
- Is a complex symmetric square matrix with zero diagonal diagonalizable?
- Symmetry of the tetrahedron as a subgroup of the cube
- Rotating a matrix to become symmetric
- Diagonalize real symmetric matrix
- How to solve for $L$ in $X = LL^T$?
- Showing a block matrix is SPD
- Proving symmetric matrix has positive eigenvalues
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?
For any symmetric matrix $A$, we have $$ \frac 12 (x - y)^TA(x - y) = \frac{x^TAx + y^TAy}{2} - x^TAy. $$ With that, we can conclude that your inequality will hold for all $x,y$ (with norm less than $1$) if and only if $A$ is negative semidefinite. If $A$ is positive semidefinite, then the opposite inequality will hold. In other cases, neither inequality holds for all $x,y$ (with norm less than $1$).