# Difference between revisions of "Determinant"

(→Equivalence of Definitions) |
(determinants in terms of simpler determinants) |
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Our generalized determinants also satisfy the multiplicative property | Our generalized determinants also satisfy the multiplicative property | ||

when <math>A</math> is [[associative]]. | when <math>A</math> is [[associative]]. | ||

+ | |||

+ | == Determinants in Terms of Simpler Determinants == | ||

+ | |||

+ | An <math>n\times n</math> determinant can be written in terms of <math>(n-1)\times(n-1)</math> determinants: | ||

+ | <cmath>\det(a_{n\times n})=\sum_{i=1}^n(-1)^{i+1}\det(m_i)</cmath> | ||

+ | where <math>m_i</math> is the <math>(n-1)\times(n-1)</math> matrix formed by removing the <math>1</math>st row and <math>i</math>th column from <math>a</math>: | ||

+ | <cmath>m_i=\begin{pmatrix} | ||

+ | \blacksquare & \blacksquare & ... & \blacksquare & \blacksquare & \blacksquare & ... & \blacksquare & \blacksquare\\ | ||

+ | a_{21} & a_{22} & ... & a_{2,i-1} & \blacksquare & a_{2,i+1} & ... & a_{2,n-1} & a_{2n}\\ | ||

+ | a_{31} & a_{32} & ... & a_{3,i-1} & \blacksquare & a_{3,i+1} & ... & a_{3,n-1} & a_{3n}\\ | ||

+ | &&&&\vdots&&&&\\ | ||

+ | a_{n1} & a_{n2} & ... & a_{n,i-1} & \blacksquare & a_{n,i+1} & ... & a_{n,n-1} & a_{nn} | ||

+ | \end{pmatrix}=\begin{pmatrix} | ||

+ | a_{21} & ... & a_{2,i-1} & a_{2,i+1} & ... & a_{2n}\\ | ||

+ | a_{31} & ... & a_{3,i-1} & a_{3,i+1} & ... & a_{3n}\\ | ||

+ | &&\vdots&\vdots&&&\\ | ||

+ | a_{n1} & ... & a_{n,i-1} & a_{n,i+1} & ... & a_{nn} | ||

+ | \end{pmatrix}</cmath> | ||

+ | |||

+ | This makes it easy to see why the determinant of an <math>n\times n</math> matrix <math>a</math> is the sum of the diagonals labeled <math>+</math>, minus the sum of the diagonals labeled <math>-</math>, where "diagonal" means the product of the terms along it: | ||

+ | <center><asy> | ||

+ | size(300); | ||

+ | draw(arc((3.3,-1.5), 4, 155, 205)); | ||

+ | draw(arc((-0.3,-1.5), 4, -25, 25)); | ||

+ | label("$a_{11}$",(0,0)); | ||

+ | label("$a_{12}$",(1,0)); | ||

+ | label("$...$",(2,0)); | ||

+ | label("$a_{1n}$",(3,0)); | ||

+ | |||

+ | label("$a_{21}$",(0,-1)); | ||

+ | label("$a_{22}$",(1,-1)); | ||

+ | label("$...$",(2,-1)); | ||

+ | label("$a_{2n}$",(3,-1)); | ||

+ | |||

+ | label("$\vdots$",(0,-2)); | ||

+ | label("$\ddots$",(2,-2)); | ||

+ | |||

+ | label("$a_{n1}$",(0,-3)); | ||

+ | label("$a_{n2}$",(1,-3)); | ||

+ | label("$...$",(2,-3)); | ||

+ | label("$a_{nn}$",(3,-3)); | ||

+ | label("$a_{11}$",(0,0)); | ||

+ | label("$a_{12}$",(1,0)); | ||

+ | label("$...$",(2,0)); | ||

+ | label("$a_{1n}$",(3,0)); | ||

+ | |||

+ | |||

+ | label("$a_{11}$",(4,0)); | ||

+ | label("$a_{12}$",(5,0)); | ||

+ | label("$...$",(6,0)); | ||

+ | label("$a_{1n}$",(7,0)); | ||

+ | |||

+ | label("$a_{21}$",(4,-1)); | ||

+ | label("$a_{22}$",(5,-1)); | ||

+ | label("$...$",(6,-1)); | ||

+ | label("$a_{2n}$",(7,-1)); | ||

+ | |||

+ | label("$\vdots$",(4,-2)); | ||

+ | label("$\ddots$",(6,-2)); | ||

+ | |||

+ | label("$a_{n1}$",(4,-3)); | ||

+ | label("$a_{n2}$",(5,-3)); | ||

+ | label("$...$",(6,-3)); | ||

+ | label("$a_{nn}$",(7,-3)); | ||

+ | |||

+ | for (int i=0; i<4; ++i) | ||

+ | { | ||

+ | draw((i-0.2,0.2)--(i+3-0.2,-3+0.2)); | ||

+ | label("$+$",(i-0.4,0.4)); | ||

+ | } | ||

+ | for (int i=0; i<4; ++i) | ||

+ | { | ||

+ | draw((i+0.2+4,0.2)--(i-3+4+0.2,-3+0.2)); | ||

+ | label("$-$",(i+0.4+4,0.4)); | ||

+ | } | ||

+ | </asy></center> | ||

== Matrix Determinants are Multiplicative == | == Matrix Determinants are Multiplicative == | ||

Line 115: | Line 191: | ||

as the characteristic polynomial does not change under | as the characteristic polynomial does not change under | ||

[[automorphism]]s of <math>A</math> that fix <math>F</math>. Our two definitions | [[automorphism]]s of <math>A</math> that fix <math>F</math>. Our two definitions | ||

− | are therefore equivalent. <math>\ | + | are therefore equivalent. <math>\blacksquare</math> |

== References == | == References == |

## Latest revision as of 00:31, 9 May 2020

The **determinant** is an important notion in linear algebra.

For an matrix , the determinant is defined by the sum where is the set of all permutations on the set , and is the parity of the permutation .

For example, the determinant of a matrix is .

This quantity may seem unwieldy, but surprisingly, it is multiplicative. That is, for any matrices (over the same commutative field),

More generally, if is a commutative field and is an element of a (strictly power associative) -dimensional -algebra , then the determinant of is times the constant term of the characteristic polynomial of .

Our generalized determinants also satisfy the multiplicative property when is associative.

## Contents

## Determinants in Terms of Simpler Determinants

An determinant can be written in terms of determinants: where is the matrix formed by removing the st row and th column from :

This makes it easy to see why the determinant of an matrix is the sum of the diagonals labeled , minus the sum of the diagonals labeled , where "diagonal" means the product of the terms along it:

## Matrix Determinants are Multiplicative

In this section we prove that the determinant as defined for matrices is multiplicative.

We first note that from rearrangements of terms. If we let , we then have

On the other hand, where is the set , and is the set of functions mapping into itself.

From equation (1), it thus suffices to show that if is not a permutation on , then

To this end, suppose that is not a permutation. Then there exist distinct integers such that . Let be the permutation on that transposes and while fixing everything else. Then as the latter product is the same as the former, with two terms switched. On the other hand is an odd permutation, so Since , we can partition the elements of into pairs for which the equation above holds. Equation 2 then follows, and we are done.

## Equivalence of Definitions

We now prove that our two definitions are equivalent. We first note that the definitions coincide in the case of upper-triangular matrices, as each entry in the diagonal of an upper-triangular matrix corresponds to a (generalized) eigenvalue of the matrix.

We now use the fact that every element of is similar to an upper triangular matrix; that is, there exists an upper triangular matrix and an invertible matrix such that Writing for our specialized determinant for matrices and for our generalized definition with the characteristic polynomial, we have as the characteristic polynomial does not change under automorphisms of that fix . Our two definitions are therefore equivalent.

## References

- Garibaldi, Skip, "The Characteristic Polynomial and Determinant Are Not Ad Hoc Constructions".
*American Mathematical Monthly***111**(2004), no. 9, p. 761, Nov. 2004. Preprint