Mixing
Determinant definition from recursion
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I know this definition of Principle of Recursive Definition from Munkres Topology, Ch 1:
Principle of Recursive Definition: Let $A$ be a set; let $a_0$ be an element of $A$. Suppose $rho$ is a function that assigns, to each function $f$ mapping a nonempty section of the positive integers into $A$, and element of $A$. Then there exists a unique function $$h:mathbb Z_+to A$$ such that $$h_1=a_0, text{ and for } i>1,, h(i)=rho(h|{1,ldots,i-1})tag{*}$$ The formula $(*)$ is called a recursion formula for $h$. It specifies $h(1)$ and it expresses the value of $h$ at $i>1$ in terms of the values of $h$ for positive integer less than $i$.
Now, I was reading Artin's Algebra, and he says that they define determinant recursively: $$det:Bbb R^{ntimes n}to mathbb R.$$
The determinant of $1times 1$ matrix $A=[a]$ is $det A=a$, and for higher dimensions, $$det A=sum_nupm a_{nu1}det A_{nu1}.$$How can I apply principle of recursive definition to arrive at this definition of determinant?
definition determinant recursion
asked Jan 18 at 7:17
SilentSilent
2,84132152
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add a comment |
$begingroup$
I know this definition of Principle of Recursive Definition from Munkres Topology, Ch 1:
Principle of Recursive Definition: Let $A$ be a set; let $a_0$ be an element of $A$. Suppose $rho$ is a function that assigns, to each function $f$ mapping a nonempty section of the positive integers into $A$, and element of $A$. Then there exists a unique function $$h:mathbb Z_+to A$$ such that $$h_1=a_0, text{ and for } i>1,, h(i)=rho(h|{1,ldots,i-1})tag{*}$$ The formula $(*)$ is called a recursion formula for $h$. It specifies $h(1)$ and it expresses the value of $h$ at $i>1$ in terms of the values of $h$ for positive integer less than $i$.
Now, I was reading Artin's Algebra, and he says that they define determinant recursively: $$det:Bbb R^{ntimes n}to mathbb R.$$
The determinant of $1times 1$ matrix $A=[a]$ is $det A=a$, and for higher dimensions, $$det A=sum_nupm a_{nu1}det A_{nu1}.$$How can I apply principle of recursive definition to arrive at this definition of determinant?
definition determinant recursion
asked Jan 18 at 7:17
SilentSilent
2,84132152
$endgroup$
add a comment |
$begingroup$
I know this definition of Principle of Recursive Definition from Munkres Topology, Ch 1:
Principle of Recursive Definition: Let $A$ be a set; let $a_0$ be an element of $A$. Suppose $rho$ is a function that assigns, to each function $f$ mapping a nonempty section of the positive integers into $A$, and element of $A$. Then there exists a unique function $$h:mathbb Z_+to A$$ such that $$h_1=a_0, text{ and for } i>1,, h(i)=rho(h|{1,ldots,i-1})tag{*}$$ The formula $(*)$ is called a recursion formula for $h$. It specifies $h(1)$ and it expresses the value of $h$ at $i>1$ in terms of the values of $h$ for positive integer less than $i$.
Now, I was reading Artin's Algebra, and he says that they define determinant recursively: $$det:Bbb R^{ntimes n}to mathbb R.$$
The determinant of $1times 1$ matrix $A=[a]$ is $det A=a$, and for higher dimensions, $$det A=sum_nupm a_{nu1}det A_{nu1}.$$How can I apply principle of recursive definition to arrive at this definition of determinant?
definition determinant recursion
asked Jan 18 at 7:17
SilentSilent
2,84132152
$endgroup$
I know this definition of Principle of Recursive Definition from Munkres Topology, Ch 1:
Principle of Recursive Definition: Let $A$ be a set; let $a_0$ be an element of $A$. Suppose $rho$ is a function that assigns, to each function $f$ mapping a nonempty section of the positive integers into $A$, and element of $A$. Then there exists a unique function $$h:mathbb Z_+to A$$ such that $$h_1=a_0, text{ and for } i>1,, h(i)=rho(h|{1,ldots,i-1})tag{*}$$ The formula $(*)$ is called a recursion formula for $h$. It specifies $h(1)$ and it expresses the value of $h$ at $i>1$ in terms of the values of $h$ for positive integer less than $i$.
Now, I was reading Artin's Algebra, and he says that they define determinant recursively: $$det:Bbb R^{ntimes n}to mathbb R.$$
The determinant of $1times 1$ matrix $A=[a]$ is $det A=a$, and for higher dimensions, $$det A=sum_nupm a_{nu1}det A_{nu1}.$$How can I apply principle of recursive definition to arrive at this definition of determinant?
definition determinant recursion
definition determinant recursion
asked Jan 18 at 7:17
SilentSilent
2,84132152
asked Jan 18 at 7:17
SilentSilent
2,84132152
asked Jan 18 at 7:17
SilentSilent
2,84132152
asked Jan 18 at 7:17
SilentSilent
2,84132152
asked Jan 18 at 7:17
SilentSilent
2,84132152
2,84132152
add a comment |
add a comment |
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