Principal axis of a matrix
$begingroup$
I try to find the definition of the main axis of a matrix.
I saw this phrase in some exercise:
Let $A$ be a positive matrix, $f:Glongrightarrow mathbb{R}$ a smooth function, $G$ an open set in $mathbb{R}^n$. I need to find the orthogonal coordinate transformation $y=Px$ such that the main axis on $y$'s coordinates will be the principle axis of $A$.
The book says to diagonalize $A$: $PAP^t=D$ and to choose $P$ to be the transformation.
What is the definition of principle axis of matrix?
thanks.
linear-algebra matrices diagonalization
edited May 29 '15 at 16:11
abel
26.6k12048
asked Apr 9 '13 at 20:59
user56714user56714
261211
$endgroup$
add a comment |
$begingroup$
I try to find the definition of the main axis of a matrix.
I saw this phrase in some exercise:
Let $A$ be a positive matrix, $f:Glongrightarrow mathbb{R}$ a smooth function, $G$ an open set in $mathbb{R}^n$. I need to find the orthogonal coordinate transformation $y=Px$ such that the main axis on $y$'s coordinates will be the principle axis of $A$.
The book says to diagonalize $A$: $PAP^t=D$ and to choose $P$ to be the transformation.
What is the definition of principle axis of matrix?
thanks.
linear-algebra matrices diagonalization
edited May 29 '15 at 16:11
abel
26.6k12048
asked Apr 9 '13 at 20:59
user56714user56714
261211
$endgroup$
$begingroup$
the principal axis of a matrix $A$ are the basis vectors so that $A$ has a diagonal matrix representation.
$endgroup$
– abel
May 29 '15 at 16:13
1
$begingroup$
I'm troubled by the "the" in this statement. Is this an English translation issue? Do we mean a principal [NOT principle] axis?
$endgroup$
– Ted Shifrin
Jan 31 '18 at 23:18
add a comment |
$begingroup$
I try to find the definition of the main axis of a matrix.
I saw this phrase in some exercise:
Let $A$ be a positive matrix, $f:Glongrightarrow mathbb{R}$ a smooth function, $G$ an open set in $mathbb{R}^n$. I need to find the orthogonal coordinate transformation $y=Px$ such that the main axis on $y$'s coordinates will be the principle axis of $A$.
The book says to diagonalize $A$: $PAP^t=D$ and to choose $P$ to be the transformation.
What is the definition of principle axis of matrix?
thanks.
linear-algebra matrices diagonalization
edited May 29 '15 at 16:11
abel
26.6k12048
asked Apr 9 '13 at 20:59
user56714user56714
261211
$endgroup$
I try to find the definition of the main axis of a matrix.
I saw this phrase in some exercise:
Let $A$ be a positive matrix, $f:Glongrightarrow mathbb{R}$ a smooth function, $G$ an open set in $mathbb{R}^n$. I need to find the orthogonal coordinate transformation $y=Px$ such that the main axis on $y$'s coordinates will be the principle axis of $A$.
The book says to diagonalize $A$: $PAP^t=D$ and to choose $P$ to be the transformation.
What is the definition of principle axis of matrix?
thanks.
linear-algebra matrices diagonalization
linear-algebra matrices diagonalization
edited May 29 '15 at 16:11
abel
26.6k12048
asked Apr 9 '13 at 20:59
user56714user56714
261211
edited May 29 '15 at 16:11
abel
26.6k12048
asked Apr 9 '13 at 20:59
user56714user56714
261211
edited May 29 '15 at 16:11
abel
26.6k12048
edited May 29 '15 at 16:11
abel
26.6k12048
edited May 29 '15 at 16:11
abel
26.6k12048
26.6k12048
asked Apr 9 '13 at 20:59
user56714user56714
261211
asked Apr 9 '13 at 20:59
user56714user56714
261211
asked Apr 9 '13 at 20:59
user56714user56714
261211
261211
$begingroup$
the principal axis of a matrix $A$ are the basis vectors so that $A$ has a diagonal matrix representation.
$endgroup$
– abel
May 29 '15 at 16:13
1
$begingroup$
I'm troubled by the "the" in this statement. Is this an English translation issue? Do we mean a principal [NOT principle] axis?
$endgroup$
– Ted Shifrin
Jan 31 '18 at 23:18
add a comment |
$begingroup$
the principal axis of a matrix $A$ are the basis vectors so that $A$ has a diagonal matrix representation.
$endgroup$
– abel
May 29 '15 at 16:13
1
$begingroup$
I'm troubled by the "the" in this statement. Is this an English translation issue? Do we mean a principal [NOT principle] axis?
$endgroup$
– Ted Shifrin
Jan 31 '18 at 23:18
$begingroup$
the principal axis of a matrix $A$ are the basis vectors so that $A$ has a diagonal matrix representation.
$endgroup$
– abel
May 29 '15 at 16:13
$begingroup$
the principal axis of a matrix $A$ are the basis vectors so that $A$ has a diagonal matrix representation.
$endgroup$
– abel
May 29 '15 at 16:13
1
1
$begingroup$
I'm troubled by the "the" in this statement. Is this an English translation issue? Do we mean a principal [NOT principle] axis?
$endgroup$
– Ted Shifrin
Jan 31 '18 at 23:18
$begingroup$
I'm troubled by the "the" in this statement. Is this an English translation issue? Do we mean a principal [NOT principle] axis?
$endgroup$
– Ted Shifrin
Jan 31 '18 at 23:18
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Often, principal axes of a matrix refer to its eigenvectors. With this diagonalization, $P$ is the matrix of eigenvectors.
answered Apr 9 '13 at 21:01
Ross B.Ross B.
1,657516
$endgroup$
$begingroup$
when you diagnolize matrix with orthogonal matrix p,p doesn't have to contain eigenvectors(grahm schmidt process might change them)
$endgroup$
– user56714
Apr 9 '13 at 21:17
1
$begingroup$
actually, you can only diagonalize $A$ using orthogonal matrix $P$ when $A$ is symmetric (why?). It follows that $P$ is always composed of eigenvector columns.
$endgroup$
– Quang Hoang
Sep 10 '15 at 11:09
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Often, principal axes of a matrix refer to its eigenvectors. With this diagonalization, $P$ is the matrix of eigenvectors.
answered Apr 9 '13 at 21:01
Ross B.Ross B.
1,657516
$endgroup$
$begingroup$
when you diagnolize matrix with orthogonal matrix p,p doesn't have to contain eigenvectors(grahm schmidt process might change them)
$endgroup$
– user56714
Apr 9 '13 at 21:17
1
$begingroup$
actually, you can only diagonalize $A$ using orthogonal matrix $P$ when $A$ is symmetric (why?). It follows that $P$ is always composed of eigenvector columns.
$endgroup$
– Quang Hoang
Sep 10 '15 at 11:09
add a comment |
$begingroup$
Often, principal axes of a matrix refer to its eigenvectors. With this diagonalization, $P$ is the matrix of eigenvectors.
answered Apr 9 '13 at 21:01
Ross B.Ross B.
1,657516
$endgroup$
$begingroup$
when you diagnolize matrix with orthogonal matrix p,p doesn't have to contain eigenvectors(grahm schmidt process might change them)
$endgroup$
– user56714
Apr 9 '13 at 21:17
1
$begingroup$
actually, you can only diagonalize $A$ using orthogonal matrix $P$ when $A$ is symmetric (why?). It follows that $P$ is always composed of eigenvector columns.
$endgroup$
– Quang Hoang
Sep 10 '15 at 11:09
add a comment |
$begingroup$
Often, principal axes of a matrix refer to its eigenvectors. With this diagonalization, $P$ is the matrix of eigenvectors.
answered Apr 9 '13 at 21:01
Ross B.Ross B.
1,657516
$endgroup$
Often, principal axes of a matrix refer to its eigenvectors. With this diagonalization, $P$ is the matrix of eigenvectors.
answered Apr 9 '13 at 21:01
Ross B.Ross B.
1,657516
answered Apr 9 '13 at 21:01
Ross B.Ross B.
1,657516
answered Apr 9 '13 at 21:01
Ross B.Ross B.
1,657516
answered Apr 9 '13 at 21:01
Ross B.Ross B.
1,657516
1,657516
$begingroup$
when you diagnolize matrix with orthogonal matrix p,p doesn't have to contain eigenvectors(grahm schmidt process might change them)
$endgroup$
– user56714
Apr 9 '13 at 21:17
1
$begingroup$
actually, you can only diagonalize $A$ using orthogonal matrix $P$ when $A$ is symmetric (why?). It follows that $P$ is always composed of eigenvector columns.
$endgroup$
– Quang Hoang
Sep 10 '15 at 11:09
add a comment |
$begingroup$
when you diagnolize matrix with orthogonal matrix p,p doesn't have to contain eigenvectors(grahm schmidt process might change them)
$endgroup$
– user56714
Apr 9 '13 at 21:17
1
$begingroup$
actually, you can only diagonalize $A$ using orthogonal matrix $P$ when $A$ is symmetric (why?). It follows that $P$ is always composed of eigenvector columns.
$endgroup$
– Quang Hoang
Sep 10 '15 at 11:09
$begingroup$
when you diagnolize matrix with orthogonal matrix p,p doesn't have to contain eigenvectors(grahm schmidt process might change them)
$endgroup$
– user56714
Apr 9 '13 at 21:17
$begingroup$
when you diagnolize matrix with orthogonal matrix p,p doesn't have to contain eigenvectors(grahm schmidt process might change them)
$endgroup$
– user56714
Apr 9 '13 at 21:17
1
1
$begingroup$
actually, you can only diagonalize $A$ using orthogonal matrix $P$ when $A$ is symmetric (why?). It follows that $P$ is always composed of eigenvector columns.
$endgroup$
– Quang Hoang
Sep 10 '15 at 11:09
$begingroup$
actually, you can only diagonalize $A$ using orthogonal matrix $P$ when $A$ is symmetric (why?). It follows that $P$ is always composed of eigenvector columns.
$endgroup$
– Quang Hoang
Sep 10 '15 at 11:09
add a comment |
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$begingroup$
the principal axis of a matrix $A$ are the basis vectors so that $A$ has a diagonal matrix representation.
$endgroup$
– abel
May 29 '15 at 16:13
1
$begingroup$
I'm troubled by the "the" in this statement. Is this an English translation issue? Do we mean a principal [NOT principle] axis?
$endgroup$
– Ted Shifrin
Jan 31 '18 at 23:18