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Linear Algebra With Applications
Found in: Page 234
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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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 AT.

L=RT, Q=PT and A=LQ

See the step by step solution

Step by Step Solution

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 det A=det ATand therefore, AT is also invertible. Now we have PR factorization for ,i.e.

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


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

Hence, L=RT, Q=PT and A=LQ.

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