Problem

Construct a spectral decomposition of $A$ from Ex…

07:53

Problem 32 Medium Difficulty

Suppose $A=P R P^{-1},$ where $P$ is orthogonal and $R$ is upper triangular. Show that if $A$ is symmetric, then $R$ is symmetric and hence is actually a diagonal matrix.

Name: suppose ap r p 1 where p is orthogonal and r is upper triangular show that if a is symmetric then r
Uploaded: 2020-07-15
Duration: 2 min 52 s
Channel: Numerade
Description: Suppose $A=P R P^{-1},$ where $P$ is orthogonal and $R$ is upper triangular. Show that if $A$ is symmetric, then $R$ is symmetric and hence is actually a diagonal matrix.

Answer

Hence $R^{T}=\left(P^{T} A P\right)^{T}=P^{T} A^{T} P^{T T}=P^{T} A P=R,$ which shows that $\mathrm{R}$ is symmetric. since $\mathrm{R}$ is also upper triangular, its entries above the diagonal must be zeros to match the zeros below the diagonal. Thus $\mathrm{R}$ is a diagonal matrix.

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Algebra

Linear Algebra and Its Applications

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 1

Diagonalization of Symmetric Matrices

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Video Transcript

we were asked to prove of fear. Um, belt special kinds of nature sees were given that a matrix a is equal to P R P in verse where p is an orthogonal matrix in ours and upper triangular matrix were asked to show that if the Matrix a is symmetric so that he transpose equals a then it follows that are symmetric and hence is actually a diagonal matrix. So we're going to also suppose a is symmetric. This means that definition a transpose equals a We want to show that our is symmetric. So we have that our transpose is equal to by rearranging using multiplication and in verses of matrices we obtain p inverse a key transpose and by our thought banality of p. This is equal to p transpose a p transpose which by the properties of transposition, is a transposed in the middle. And then we have p transposing the left and p transposed transposed on the right and again but the properties of transposition we had that this is going to be p transpose a transpose and then simply he and this is equal to since is symmetric and his peers orthogonal p in verse AP if we recognize using matrix multiplication and in verses is the Matrix are so it follows that our transpose equals R and R is symmetric by definition, and since ours upper triangular and symmetric, it follows that are is a diagonal matrix.

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