Short Answer

Consider an invertible n ×n matrix A. Can you write A as A = LQ, where L is a lower triangular matrix and Q is orthogonal? Hint: Consider the QR factorization of $A^{T}$ .

$L = R^{T}, Q = P^{T} and A = L Q$

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Step by Step Solution

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Step 1: L is a lower triangular matrix.

Given that A is an n ×n invertible matrix. We claimed that there exists lower triangular matrix L and an orthogonal matrix Q such that A=LQ.

Since $d e t A = d e t A^{T}$ and therefore, $A^{T}$ is also invertible. Now we have PR factorization for ,i.e.

There exists upper triangular matrix R and an orthogonal matrix P such that

$A^{T} = P R = {(A^{T})}^{T} = {(P R)}^{T} = A \Rightarrow R^{T} P^{T} = {(A B)}^{T} = B^{T} A^{T}$

By observing in the above factorization. Since R is an upper triangular matrix therefore, will be lower triangular matrix. Also, the matrix $P^{T}$ is orthogonal and hence it has a factorization for A as lower triangular matrix and an orthonormal matrix.

Hence, $L = R^{T}, Q = P^{T} and A = L Q$ .

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Linear Algebra With Applications

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Short Answer

Consider an invertible n ×n matrix A. Can you write A as A = LQ, where L is a lower triangular matrix and Q is orthogonal? Hint: Consider the QR factorization of $A^{T}$ .

Step by Step Solution

Step 1: L is a lower triangular matrix.

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Linear Algebra With Applications

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Consider an invertible n ×n matrix A. Can you write A as A = LQ, where L is a lower triangular matrix and Q is orthogonal? Hint: Consider the QR factorization of AT.

Step by Step Solution

Step 1: L is a lower triangular matrix.

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Consider an invertible n ×n matrix A. Can you write A as A = LQ, where L is a lower triangular matrix and Q is orthogonal? Hint: Consider the QR factorization of $A^{T}$ .