How to prove the following question by using determinant?
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Let $A$ be a $mtimes m$ nonsingular matrix. Show that $A$ has an LU factorization if and only if for each $k$ with $1leq kleq m$, the upper-left $ktimes k$ block $ A_{1:k,1:k}$ is nonsingular. The row operations of Gaussian elimination leave the determinants $det( A_{1:k,1:k} )$ unchanged. Prove that this LU factorization is unique.
linear-algebra
asked Dec 18 '18 at 13:38
KristyKristy
204
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add a comment |
$begingroup$
Let $A$ be a $mtimes m$ nonsingular matrix. Show that $A$ has an LU factorization if and only if for each $k$ with $1leq kleq m$, the upper-left $ktimes k$ block $ A_{1:k,1:k}$ is nonsingular. The row operations of Gaussian elimination leave the determinants $det( A_{1:k,1:k} )$ unchanged. Prove that this LU factorization is unique.
linear-algebra
asked Dec 18 '18 at 13:38
KristyKristy
204
$endgroup$
$begingroup$
Closely related Questions have been asked a number of times here, but in a quick search I did not find one that highlights the uniqueness of the LU factorization, and perhaps worse from the perspective of wanting give an exact duplicate, the Answers to those (concerning existence and nonzero principal minors) tend to be provided as hints rather than complete arguments. I'll look further, though you can find the claim roughly stated in the Wikipedia article.
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– hardmath
Dec 20 '18 at 6:48
add a comment |
$begingroup$
Let $A$ be a $mtimes m$ nonsingular matrix. Show that $A$ has an LU factorization if and only if for each $k$ with $1leq kleq m$, the upper-left $ktimes k$ block $ A_{1:k,1:k}$ is nonsingular. The row operations of Gaussian elimination leave the determinants $det( A_{1:k,1:k} )$ unchanged. Prove that this LU factorization is unique.
linear-algebra
asked Dec 18 '18 at 13:38
KristyKristy
204
$endgroup$
Let $A$ be a $mtimes m$ nonsingular matrix. Show that $A$ has an LU factorization if and only if for each $k$ with $1leq kleq m$, the upper-left $ktimes k$ block $ A_{1:k,1:k}$ is nonsingular. The row operations of Gaussian elimination leave the determinants $det( A_{1:k,1:k} )$ unchanged. Prove that this LU factorization is unique.
linear-algebra
linear-algebra
asked Dec 18 '18 at 13:38
KristyKristy
204
asked Dec 18 '18 at 13:38
KristyKristy
204
asked Dec 18 '18 at 13:38
KristyKristy
204
asked Dec 18 '18 at 13:38
KristyKristy
204
asked Dec 18 '18 at 13:38
KristyKristy
204
204
$begingroup$
Closely related Questions have been asked a number of times here, but in a quick search I did not find one that highlights the uniqueness of the LU factorization, and perhaps worse from the perspective of wanting give an exact duplicate, the Answers to those (concerning existence and nonzero principal minors) tend to be provided as hints rather than complete arguments. I'll look further, though you can find the claim roughly stated in the Wikipedia article.
$endgroup$
– hardmath
Dec 20 '18 at 6:48
add a comment |
$begingroup$
Closely related Questions have been asked a number of times here, but in a quick search I did not find one that highlights the uniqueness of the LU factorization, and perhaps worse from the perspective of wanting give an exact duplicate, the Answers to those (concerning existence and nonzero principal minors) tend to be provided as hints rather than complete arguments. I'll look further, though you can find the claim roughly stated in the Wikipedia article.
$endgroup$
– hardmath
Dec 20 '18 at 6:48
$begingroup$
Closely related Questions have been asked a number of times here, but in a quick search I did not find one that highlights the uniqueness of the LU factorization, and perhaps worse from the perspective of wanting give an exact duplicate, the Answers to those (concerning existence and nonzero principal minors) tend to be provided as hints rather than complete arguments. I'll look further, though you can find the claim roughly stated in the Wikipedia article.
$endgroup$
– hardmath
Dec 20 '18 at 6:48
$begingroup$
Closely related Questions have been asked a number of times here, but in a quick search I did not find one that highlights the uniqueness of the LU factorization, and perhaps worse from the perspective of wanting give an exact duplicate, the Answers to those (concerning existence and nonzero principal minors) tend to be provided as hints rather than complete arguments. I'll look further, though you can find the claim roughly stated in the Wikipedia article.
$endgroup$
– hardmath
Dec 20 '18 at 6:48
add a comment |
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$begingroup$
Closely related Questions have been asked a number of times here, but in a quick search I did not find one that highlights the uniqueness of the LU factorization, and perhaps worse from the perspective of wanting give an exact duplicate, the Answers to those (concerning existence and nonzero principal minors) tend to be provided as hints rather than complete arguments. I'll look further, though you can find the claim roughly stated in the Wikipedia article.
$endgroup$
– hardmath
Dec 20 '18 at 6:48