show all steps to solution The Gram matrix of x₁,..., Xn E Rm is the n x n matrix G whereGi,j=Xi•Xj,for i, j = 1,..., n.Xn then G = BT B.For example,if B =3X1 X21 13...is the Gram matrix of x₁ =1and x2 =03TDefinition: A matrix A is positive semidefinite if it is symmetric and x Ax ≥ 0, for all x € R".(a) Show that the Gram matrix of x₁,...,xn € Rm is positive semidefinite.(b) Show that an n x n positive semidefinite matrix A is the Gram matrix of n vectors in R".use the orthogonal diagonalization of A

Answered: Image /qna-images/answer/ada36207-3a75-4293-93ac-23dd18fb58ce.jpg

(a) Show that the Gram matrix of x₁,...,xn € Rm is positive semidefinite. (b) Show that an n x n positive semidefinite matrix A is the Gram matrix of n vectors in R". use the orthogonal diagonalization of A

(a) Show that the Gram matrix of x₁,...,xn € Rm is positive semidefinite. (b) Show that an n x n positive semidefinite matrix A is the Gram matrix of n vectors in R". use the orthogonal diagonalization of A

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 23EQ

See similar textbooks

Related questions

Q: Find T' (X). f(x) = (x²-3) (x²+7) f'(x) = (Type an exact answer.)

Q: Determine whether the statement is true or false. If lim_an = 0, then an is convergent. n → ∞ False.…

Q: Let f, g be differentiable functions on [a, b] and their derivatives f', g' be integrable on [a, b].…

Q: The following set S of vectors spans R4. Find a subset BCS that is a basis for R4. S = 8 -3 Number…

Q: Replace ? with an expression that will make the equation valid. d (2 − 3x²) 7 = 7 (2-3x²) 6 ? dx

A: Differentiate using power rule and chain rule of differentiation. Power Rule of differentiation:…

Q: Find f'(x). f(x) = X x - 39 f'(x) = (Type an exact answer.)

Q: * dx Using Beta and Gamma Evaluate 5*

Q: x dx Using Beta and Gamma Evaluate 5*

Q: Give an example showing that the union of two subspaces of a vector space V is not necessarily a…

Q: Find a sequence of elementary matrices whose product is the given nonsingular matrix.

Q: 5. dr de (re) tan 0. 2011

Q: THEOREM OF THE GENERAL SOLUTION. Let x1(t),…,xn(t) be n linearly independent solutions of the system…

A: The given information is that we have a system of linear differential equations of the form X' =…

Q: The following data matrix includes 8 data points each having 2 dimensions X(1) and X(2): X(1) X(2) 1…

A: Since you have posted multiple questions, we provide the solution only to the first question as per…

Q: let AABC be a triangle S². Let the dual point C'ES² of C be defined by the following three…

A: Given: △ABC is a triangle S2 (the two-dimensional sphere). The dual point C'∈S2 of C is described by…

Q: Given the matrix A = Values of a: a a 5 3 97 -6 7 2 2 a find all values of a that make |A| = 0. Give…

Q: ne triple scalar product to find the volume of the parallelepiped having adjacent edges U, V, and w.…

A: Introduction: A parallelepiped is a three-dimensional figure formed by six parallelograms. Any of…

Q: Define the following relations on A = {1,2,3,4}. Answer each of the following questions by listing…

Q: For Problems 11-14, determine orthogonal bases for rowspace(A) and colspace(A). 11. A = - [4 12. A =…

Q: Let A cos 0 - sinᎾ sin 0 cos (a) Show that A² = cos 20 sin 20 -sin 20 cos 20

Q: (a) Find the critical points of f(x, y) = x² cos(y) + y² where - ≤ y ≤. (b) Classify the critical…

A: Given: A function fx , y=x2cosy+y2. To Find: Classify the critical points as min/max/saddle points.

Q: For each nonzero rational number in the interval (0, 1), write a as p/q where p, q are integers with…

A: The answer provided below has been developed in a clear step by step manner.

Q: 1 point) Let F₁ be the kth Fourier approximation, that is the first k harmonics: F(t): F₁(t) = F₂(t)…

A: Given that ft = -2if -π ≤ t < 05if 0 ≤ t < π The Fourier series of ft is given by Fk t =…

Q: An object of mass 5 kg is released from rest 3000 m above the ground and allowed to fall under the…

A: Given that Object mass m=5kg Released from rest 3000m above Proportionality constant b=50 N-sec/m…

Q: 2) You win a lottery worth $5,000,000 in total payments, distributed to you every four months in 30…

Q: Determine whether or not each of the following integrals is improper. In a da not improper (a). (b).…

Q: Find the volume of the region in the first octant bounded by the coordinate planes, the plane y+z=2,…

Q: a. Find the most general real-valued solution to the linear system of differential equations a¹ = 11…

Q: What is the sum-of-product expansion of the following expression? x+y O O xy + xy + xy xy + xy + xy…

Q: Patches is a cat who eats cat food for breakfast. Patches requires a breakfast diet that includes at…

A: Given: The amount of protein required is at least 40g.The amount of iron required is at least…

Q: Let f RR such that f(x) = -x if x is rational and f(x) = 2x+3 if x is irrational. Prove that f is…

Q: A certain protected species normally doubles its population each month. The initial population is…

Q: ippose we have two vectors in R³, ₁ = 8 and ₂ = 6. Find two vectors, ₁ and ₂ such that v₁ and v₂ an…

A: Introduction: To find the orthogonal set of vectors for the given set of vectors you have to use the…

Q: 114 2. Suppose that in Example 2.27, 400 units of food A, 500 units of B, and 600 units of C are…

Q: Find particular solutions for the following nonhomogeneous ODEs. (a) y′′ − 3y′ + 2y = (1 + x)e^3x…

Q: The transition matrix P for a Markov chain with states 0 and 1 is given. Assume that the chain…

Q: 3. Thomas collected some local stamps and foreign stamps in the ratio 5: 4. Calvin collected as many…

Q: Replace? with an expression that will make the equation valid. d (8x +9)5 = 5(8x +9)4 ? dx The…

A: The given equation, ddx8x+95=58x+94×?. To Find : The missing place.

Q: Find the interest rate needed for the sinking fund to reach the required amount. Assume that the…

Q: 13. An mxn upper triangular matrix is one whose entries below the main diagonal are 0's (as in…

A: (.) Consider square 'n×n' upper triangular matrix whose entries below the main diagonal entries…

Q: Let R be a commutative ring with unit which is not an integral domain. Show that quotient and…

A: Question asks to show that if R is a commutative ring with a unit that is not an integral domain,…

Q: Evaluate the integral 16" (V x F). dS= (V x F). ds, where S is the portion of the surface of a…

Q: Find conditions on a, b, c, and d such that B 1 [!] and [i] 0 1 commutes with both a b }] [c d]

Q: (1) For each of the following power series, find the convergence radius: (2) Σ (b) (c) 178 √n (3η)!…

Q: Question 7 Use Simpson's Rule with n = 4 steps to estimate the integral. O 1171 315 O 1747 315 O…

Q: 1-47 Evaluate the integral. 1. f sin'x cos³x dx 3.sin³x cos'x dx 5. f cos³x sin'x dx w/2 7. ² cos³0…

A: Since you have posted a question with multiple subparts, so we will solve only the first three…

Q: Graph each constraint to find the corner points, then give the optimal solution for each objective…

A: 1. In Job A it costs $200/podcast In Job B it costs $220/podcast The podcast producer produces at…

Q: Let R be the divisibility relation () on A = {3, 5, 6, 7, 10, 20, 30). We know that (A, I) is a…

Q: In excenopse 12, solve the given system the method of example 3.35 using 12. 24₁- 42 = 1 241 +12 =२

Q: 2. Damian deposits P dollars in a POSD bank at the beginning of a year. The deposit earns an…

Q: Proposition 8.4.6. Let U CR" be open and f: U → Rm. The function f is continuously differen- afj…

A: Introduction: The Partial Derivative Theorem, also known as Clairaut's Theorem or Schwarz's Theorem,…

Question

show all steps to solution

The Gram matrix of x₁,..., Xn E Rm is the n x n matrix G where
Gi,j
=Xi•Xj,
for i, j = 1,..., n.
Xn then G = BT B.
For example,
if B =
3
X1 X2
1 13
...
is the Gram matrix of x₁ =
1
and x2 =
0
3
T
Definition: A matrix A is positive semidefinite if it is symmetric and x Ax ≥ 0, for all x € R".
(a) Show that the Gram matrix of x₁,...,xn € Rm is positive semidefinite.
(b) Show that an n x n positive semidefinite matrix A is the Gram matrix of n vectors in R".
use the orthogonal diagonalization of A

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here