Upper Triangular Matrix and Gram-Schmidt
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If $T$ is an operator on the finite dimensional vector space $V$ which has a basis $mathcal{B}$ for which the matrix representation of $T$ with this basis is upper-triangular, does it follow that $T$ represented with respect to the basis obtained from $mathcal{B}$ by Gram-Schmidt is also upper-triangular?
linear-algebra matrices inner-product-space gram-schmidt
asked Nov 13 at 20:35
user193319
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If $T$ is an operator on the finite dimensional vector space $V$ which has a basis $mathcal{B}$ for which the matrix representation of $T$ with this basis is upper-triangular, does it follow that $T$ represented with respect to the basis obtained from $mathcal{B}$ by Gram-Schmidt is also upper-triangular?
linear-algebra matrices inner-product-space gram-schmidt
asked Nov 13 at 20:35
user193319
2,2862922
Let $mathcal{C}$ be the basis obtained from $mathcal{B}$ by Gram-Schmidt. Is the change-of-basis matrix between $mathcal{B}$ and $mathcal{C}$ upper-triangular? (The answer depends somewhat on how exactly you define Gram-Schmidt.) If the answer is "yes", then the answer to your question is "yes" as well, because products and inverses of upper-triangular matrices are upper-triangular.
– darij grinberg
Nov 13 at 20:58
@darijgrinberg By Gram-Schmidt, I mean by procedure whereby one transforms a basis into an orthonormal basis. Is that what you had in mind, too?
– user193319
Nov 13 at 21:00
Yes, but there are several of these procedures. In your version, is the $j$-th vector of $mathcal{C}$ a linear combination of the first $j$ vectors of $mathcal{B}$ ? If so, the answer is "yes".
– darij grinberg
Nov 13 at 21:02
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
If $T$ is an operator on the finite dimensional vector space $V$ which has a basis $mathcal{B}$ for which the matrix representation of $T$ with this basis is upper-triangular, does it follow that $T$ represented with respect to the basis obtained from $mathcal{B}$ by Gram-Schmidt is also upper-triangular?
linear-algebra matrices inner-product-space gram-schmidt
asked Nov 13 at 20:35
user193319
2,2862922
If $T$ is an operator on the finite dimensional vector space $V$ which has a basis $mathcal{B}$ for which the matrix representation of $T$ with this basis is upper-triangular, does it follow that $T$ represented with respect to the basis obtained from $mathcal{B}$ by Gram-Schmidt is also upper-triangular?
linear-algebra matrices inner-product-space gram-schmidt
linear-algebra matrices inner-product-space gram-schmidt
asked Nov 13 at 20:35
user193319
2,2862922
asked Nov 13 at 20:35
user193319
2,2862922
asked Nov 13 at 20:35
user193319
2,2862922
asked Nov 13 at 20:35
user193319
2,2862922
asked Nov 13 at 20:35
user193319
2,2862922
2,2862922
Let $mathcal{C}$ be the basis obtained from $mathcal{B}$ by Gram-Schmidt. Is the change-of-basis matrix between $mathcal{B}$ and $mathcal{C}$ upper-triangular? (The answer depends somewhat on how exactly you define Gram-Schmidt.) If the answer is "yes", then the answer to your question is "yes" as well, because products and inverses of upper-triangular matrices are upper-triangular.
– darij grinberg
Nov 13 at 20:58
@darijgrinberg By Gram-Schmidt, I mean by procedure whereby one transforms a basis into an orthonormal basis. Is that what you had in mind, too?
– user193319
Nov 13 at 21:00
Yes, but there are several of these procedures. In your version, is the $j$-th vector of $mathcal{C}$ a linear combination of the first $j$ vectors of $mathcal{B}$ ? If so, the answer is "yes".
– darij grinberg
Nov 13 at 21:02
add a comment |
Let $mathcal{C}$ be the basis obtained from $mathcal{B}$ by Gram-Schmidt. Is the change-of-basis matrix between $mathcal{B}$ and $mathcal{C}$ upper-triangular? (The answer depends somewhat on how exactly you define Gram-Schmidt.) If the answer is "yes", then the answer to your question is "yes" as well, because products and inverses of upper-triangular matrices are upper-triangular.
– darij grinberg
Nov 13 at 20:58
@darijgrinberg By Gram-Schmidt, I mean by procedure whereby one transforms a basis into an orthonormal basis. Is that what you had in mind, too?
– user193319
Nov 13 at 21:00
Yes, but there are several of these procedures. In your version, is the $j$-th vector of $mathcal{C}$ a linear combination of the first $j$ vectors of $mathcal{B}$ ? If so, the answer is "yes".
– darij grinberg
Nov 13 at 21:02
Let $mathcal{C}$ be the basis obtained from $mathcal{B}$ by Gram-Schmidt. Is the change-of-basis matrix between $mathcal{B}$ and $mathcal{C}$ upper-triangular? (The answer depends somewhat on how exactly you define Gram-Schmidt.) If the answer is "yes", then the answer to your question is "yes" as well, because products and inverses of upper-triangular matrices are upper-triangular.
– darij grinberg
Nov 13 at 20:58
Let $mathcal{C}$ be the basis obtained from $mathcal{B}$ by Gram-Schmidt. Is the change-of-basis matrix between $mathcal{B}$ and $mathcal{C}$ upper-triangular? (The answer depends somewhat on how exactly you define Gram-Schmidt.) If the answer is "yes", then the answer to your question is "yes" as well, because products and inverses of upper-triangular matrices are upper-triangular.
– darij grinberg
Nov 13 at 20:58
@darijgrinberg By Gram-Schmidt, I mean by procedure whereby one transforms a basis into an orthonormal basis. Is that what you had in mind, too?
– user193319
Nov 13 at 21:00
@darijgrinberg By Gram-Schmidt, I mean by procedure whereby one transforms a basis into an orthonormal basis. Is that what you had in mind, too?
– user193319
Nov 13 at 21:00
Yes, but there are several of these procedures. In your version, is the $j$-th vector of $mathcal{C}$ a linear combination of the first $j$ vectors of $mathcal{B}$ ? If so, the answer is "yes".
– darij grinberg
Nov 13 at 21:02
Yes, but there are several of these procedures. In your version, is the $j$-th vector of $mathcal{C}$ a linear combination of the first $j$ vectors of $mathcal{B}$ ? If so, the answer is "yes".
– darij grinberg
Nov 13 at 21:02
add a comment |
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Let $mathcal{C}$ be the basis obtained from $mathcal{B}$ by Gram-Schmidt. Is the change-of-basis matrix between $mathcal{B}$ and $mathcal{C}$ upper-triangular? (The answer depends somewhat on how exactly you define Gram-Schmidt.) If the answer is "yes", then the answer to your question is "yes" as well, because products and inverses of upper-triangular matrices are upper-triangular.
– darij grinberg
Nov 13 at 20:58
@darijgrinberg By Gram-Schmidt, I mean by procedure whereby one transforms a basis into an orthonormal basis. Is that what you had in mind, too?
– user193319
Nov 13 at 21:00
Yes, but there are several of these procedures. In your version, is the $j$-th vector of $mathcal{C}$ a linear combination of the first $j$ vectors of $mathcal{B}$ ? If so, the answer is "yes".
– darij grinberg
Nov 13 at 21:02