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In Exercises 21 and $22,$ matrices are $n \times n$ and vectors are in $\mathbb{R}^{n}$ . Mark each statement True or False. Justify each answer.a. The matrix of a quadratic form is a symmetric matrix.b. A quadratic form has no cross-product terms if and only if the matrix of the quadratic form is a diagonal matrix.c. The principal axes of a quadratic form $\mathbf{x}^{T} A \mathbf{x}$ are eigenvectors of $A .$d. A positive definite quadratic form $Q$ satisfies $Q(\mathbf{x}) > 0$ for all $\mathbf{x}$ in $\mathbb{R}^{n}$ .e. If the eigenvalues of a symmetric matrix $A$ are all positive, then the quadratic form $\mathbf{x}^{T} A \mathbf{x}$ is positive definite.f. A Cholesky factorization of a symmetric matrix $A$ has the form $A=R^{T} R,$ for an upper triangular matrix $R$ with positive diagonal entries.

A. trueB. trueC. trueD. trueE. falseF. true

Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 2

Quadratic Forms

Introduction to Matrices

Missouri State University

Baylor University

University of Michigan - Ann Arbor

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Okay, so in this problem, we have six statement. Um, we won't verify they are true or force. Okay, So hot A It is true because this is the definition off this demetrick for, um, the quadratic form. The metrics of the apologetic phone has to be a symmetric magics for party. They say so. So sure, Because if the metrics, it's a diagonal magics, something like this we used to battle magics for his import. Save this symmetric matrix. This attack on all metrics of quadratic form. Excellent x two times a comes x one x two um, were not content. And the cross product because equals two d one x one square pastie two x two square. So there's no cross product. ERM, like, excellent x two in this function. Ah, see is true. And this is the definition off the principal excess. Um, and the 40 He's also true because this is definition of the positive definite quadratic form and the U. S also true. So if for any symmetric matrix eight, we has has some idea values, for example, under 12 lambda and and they've off them a positive them is there exist of metrics p such that, um, peaches posed a p e closed toe a diagonal magics. I'm the one London and off the mat zero since off This are Londoners are positive them is for a knee X in our in x X Um, we defied Why to be X equals Dupee times. Why so x transpose a X equals Do why transpose peaches post a p times y and the equals two lambda one by one square purse lamb that to try to square up to London and Dwyane Square. So we already know that off this lambda eyes I'm positive. Mm Use X transpose t a x transpose X eyes, greater rico's zero and the equality hose if and only if XY causes, you know, which satisfy the definition of the positive definite and that if it's also true So we don't that, uh, Chesky off factory ization of a symmetric metric. They close to an upper triangle of metrics. R t aren't you supposed tops our So we're eyes that upper triangular matrix with a positive diagonal entrance. So basically this is equivalent. This is a government statement. A Jew a is positive, definite, which is in textbook

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