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Consider an invertible n × n matrix A. Can you write A = RQ, where R...

Consider an invertible n × n matrix A. Can you write A = RQ, where R is an upper triangular matrix and Q is orthogonal?
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Answer #1

Let A= [a​​​​​​1 a​​​​​​2 . . . a​​​​​​n] be an n×n invertible matrix. Each

a​​​​​​i is a colomn vector in R​​​​​​n. By gram Schmidt orthogonal process construct orthonormal basis (e​​​​​1,e​​​​​​2 . . .e​​​​​​n) using Coloms of A. Then Q=[e​​​​​​1 e​​​​​2 . . .e​​​​​​n] which is an orthogonal matrix. We define Matrix

R = [ r​​​​​​​i,j ] as r​​​​​i,j= e​​​​i.ajt for i<=j and 0 for i=j

R is an upper triangular matrix and Q is an orthogonal matrix we have QR=A

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