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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.

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.

Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 1

Diagonalization of Symmetric Matrices

Introduction to Matrices

McMaster University

Harvey Mudd College

Idaho State University

Lectures

01:32

In mathematics, the absolu…

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An $n \times n$ matrix $A$…

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Suppose $A$ is a symmetric…

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A real $n \times n$ matrix…

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Show that $A^{2}=P D^{2} P…

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Show that $A=P D P^{-1},$ …

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An $n \times n$ matrix $\m…

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Let $\mathbf{A}$ be a matr…

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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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