💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Get the answer to your homework problem.

Like

Report

Show that if an $n \times n$ matrix $A$ is positive definite, then there exists a positive definite matrix $B$ such that $A=B^{T} B .[\text { Hint: }$ Write $A=P D P^{T},$ with $P^{T}=P^{-1} .$ Produce a diagonal matrix $C$ such that $D=C^{T} C,$ and let $B=P C P^{T} .$ Show that $B$ works. $]$

Therefore, if the matrix $A$ is invertible $n \times n$ matrix and if $A$ is positive definite then there existspositive definite matrix $B$ such that $B^{T} B=A$

Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 2

Quadratic Forms

Introduction to Matrices

Oregon State University

University of Michigan - Ann Arbor

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

03:00

Let $A$ be an $m \times n$…

02:26

Suppose $A$ is a symmetric…

05:04

Let $A=\left[a_{i j}(t)\ri…

08:58

If $A=\left(\begin{array}{…

03:50

Prove that if $A$ is an $n…

03:09

Suppose $A$ is an $m \time…

04:04

Prove that if $A$ is an $m…

01:25

Let $A$ and $B$ be symmetr…

01:56

Let $A$ be the $2 \times 2…

01:41

Prove that if $A, B, C$ ar…

and and by a matrix A is positive. Definite. Then there exists a positive definite matrix Be such that a equals B transposed times beat Okay, to solve these to solve this problem on DDE the first thing we need to notice that we can write a as the product off product off p times t times he transposed. So this is just our more textbook that we can write down symmetric right down the the Matrix as Thea p. Times the diagonal times a diagonal make matrix and where p is ah, eyes orthogonal matrix and times p transpose with such p transport ese Exactly the reverse. All right. And the next thing we can observe that if we consider the matrix a diagonal matric see such dead the square off see is D that is just like taking the a square root Tell off Thea diagonal matrix D because scuse me because we for the diagonal matrix deep we're just taking the square would help each entry on the diagonal though you be and that is, see and we find the most matrix multiplication of such a C that is the same as the Dagenham. A 60 so that? It seemed to say, since C is also a diagonal matrix hens, that implies the transpose us. I'll see his the same matter. See, we have c transposed time. See Equals D. All right. And here we need to make a note. That is a trick. See, see, is also diagonal. Okay, The next thing is, consider as first let's define the matrix B to be product off P see transpose off p. And now we re calculate product product out Pete transpose and be transposed and b so that gives can you see transpose I suppose times p see he transports. So the first, the first records things in the inside the breakfast deep sighs take the transpose We have P c transposed and he transposed kinds. You see, Petey and scenes The numbers off p is equal to the transpose Uppity can suppose off p So that means to transpose lp times p will be the identity matrix as we have p times. See transposed times. See? Yeah, this part this part will be cancelled. Okay. I now recall that the product embassy transports and see is exactly deep. So his p times three times he transposed and this is exactly the matrix A as we assumed here. So we're that

View More Answers From This Book

Find Another Textbook

Numerade Educator

In mathematics, the absolute value or modulus |x| of a real number x is its …

Let $A$ be an $m \times n$ matrix and let $B$ be an $p \times n$ matrix. Use…

Suppose $A$ is a symmetric $n \times n$ matrix and $B$ is any $n \times m$ m…

Let $A=\left[a_{i j}(t)\right]$ be an $m \times n$ matrix function and let $…

If $A=\left(\begin{array}{cc}A_{0} & 0 \\ 0 & B_{0}\end{array}\right…

Prove that if $A$ is an $n \times p$ matrix and $D=$ $\operatorname{diag}\le…

Suppose $A$ is an $m \times n$ matrix and there exist $n \times m$ matrices …

Prove that if $A$ is an $m \times n$ matrix and $D=$ $\operatorname{diag}\le…

Let $A$ and $B$ be symmetric $n \times n$ matrices whose eigenvalues are all…

Let $A$ be the $2 \times 2$ matrix

$$A=\left[\begin{array}{ll}{a…

Prove that if $A, B, C$ are $n \times n$ matrices satisfying $B A=I_{n}$ and…

04:52

Use the terminology from Section 8.2Repeat Exercise 7 for $T=\left\{\lef…

04:23

Exercises $3-8$ refer to $\mathbb{P}_{2}$ with the inner product given by ev…

07:13

a. Rewrite the data in Example 1 with new $x$-coordinates in mean deviation …

01:29

In Exercises 9 and $10,$ mark each statement True or False. Justify each ans…

02:32

In Exercises $3-6,$ find (a) the maximum value of $Q(x)$ subject to the cons…

03:51

Find the matrix of the quadratic form. Assume $\mathbf{x}$ is in $\mathbb{R}…

14:22

Find a QR factorization of the matrix in Exercise 12

00:57

Determine which of the matrices in Exercises $1-6$ are symmetric. $$…

02:39

15:27

Let $\mathbb{P}_{3}$ have the inner product given by evaluation at $-3,-1$ $…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.