Definition:Determinant



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  • 1.2 Definition 1
  • 1.3 Definition 2
  • 1.4 Determinant of Linear Operator
  • 2 Rows and Columns
  • 3 Examples
  • 4 Also see
  • 5 Sources

    • Definition

    Determinant of Matrix

    Let $\mathbf A = \sqbrk a_n$ be a square matrixoforder $n$.

    That is, let:

    $\quad \mathbf A = \begin {bmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n} \\ \end {bmatrix}$


    Definition 1

    Let $\lambda: \N_{> 0} \to \N_{> 0}$ be a permutation on $\N_{>0}$.


    The determinant of $\mathbf A$ is defined as:

    $\ds \map \det {\mathbf A} := \sum_{\lambda} \paren {\map \sgn \lambda \prod_{k \mathop = 1}^n a_{k \map \lambda k} } = \sum_\lambda \map \sgn \lambda a_{1 \map \lambda 1} a_{2 \map \lambda 2} \cdots a_{n \map \lambda n}$

    where:

    the summation $\ds \sum_\lambda$ goes over all the $n!$ permutations of $\set {1, 2, \ldots, n}$
    $\map \sgn \lambda$ is the sign of the permutation $\lambda$.


    Definition 2

    The determinant of $\mathbf A$ is defined as follows:

    For $n = 1$, the order $1$ determinant is defined as:

    $\begin {vmatrix} a_{1 1} \end {vmatrix} = a_{1 1}$

    Thus the determinant of an order $1$ matrix is that element itself.


    For $n > 1$, the determinantoforder $n$ is defined recursively as:


    $\quad \ds \map \det {\mathbf A} := \begin {vmatrix} a_{1 1} & a_{1 2} & a_{1 3} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & a_{2 3} & \cdots & a_{2 n} \\ a_{3 1} & a_{3 2} & a_{3 3} & \cdots & a_{3 n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_{n 1} & a_{n 2} & a_{n 3} & \cdots & a_{n n} \\ \end {vmatrix} = a_{1 1} \begin {vmatrix} a_{2 2} & a_{2 3} & \cdots & a_{2 n} \\ a_{3 2} & a_{3 3} & \cdots & a_{3 n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n 2} & a_{n 3} & \cdots & a_{n n} \\ \end {vmatrix} - a_{1 2} \begin {vmatrix} a_{2 1} & a_{2 3} & \cdots & a_{2 n} \\ a_{3 1} & a_{3 3} & \cdots & a_{3 n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n 1} & a_{n 3} & \cdots & a_{n n} \\ \end {vmatrix} + \cdots + \paren {-1}^{n + 1} a_{1 n} \begin {vmatrix} a_{2 1} & a_{2 2} & \cdots & a_{2, n - 1} \\ a_{3 1} & a_{3 3} & \cdots & a_{3, n - 1} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n 1} & a_{n 3} & \cdots & a_{n, n - 1} \\ \end {vmatrix}$


    Determinant of Linear Operator

    Let $V$ be a finite-dimensional vector space over a field $K$.

    Let $A: V \to V$ be a linear operator of $V$.


    The determinant $\map \det A$ of $A$ is defined to be the determinant of any matrix of $A$ relative to some basis.


    Rows and Columns

    Row of Determinant

    Let $\mathbf D$ be a determinant.

    The rows of $\mathbf D$ are the lines of elements reading across the page.


    Column of Determinant

    Let $\mathbf D$ be a determinant.

    The columns of $\mathbf D$ are the lines of elements reading across the page.


    Examples

    Determinant of Order 1

    This is the trivial case:

    $\begin {vmatrix} a_{1 1} \end {vmatrix} = a_{1 1}$

    Thus the determinant of an order $1$ matrix is that element itself.


    Determinant of Order 2

    \(\ds \begin {vmatrix} a_{1 1} & a_{1 2} \\ a_{2 1} & a_{2 2} \end{vmatrix}\) \(=\) \(\ds \map \sgn {1, 2} a_{1 1} a_{2 2} + \map \sgn {2, 1} a_{1 2} a_{2 1}\)
    \(\ds \) \(=\) \(\ds a_{1 1} a_{2 2} - a_{1 2} a_{2 1}\)


    Determinant of Order 3

    Let:

    $\quad \map \det {\mathbf A} = \begin {vmatrix} a_{1 1} & a_{1 2} & a_{1 3} \\ a_{2 1} & a_{2 2} & a_{2 3} \\ a_{3 1} & a_{3 2} & a_{3 3} \end {vmatrix}$


    Then:

    \(\ds \map \det {\mathbf A}\) \(=\) \(\ds a_{1 1} \begin {vmatrix} a_{2 2} & a_{2 3} \\ a_{3 2} & a_{3 3} \end {vmatrix} - a_{1 2} \begin {vmatrix} a_{2 1} & a_{2 3} \\ a_{3 1} & a_{3 3} \end {vmatrix} + a_{1 3} \begin {vmatrix} a_{2 1} & a_{2 2} \\ a_{3 1} & a_{3 2} \end{vmatrix}\)
    \(\ds \) \(=\) \(\ds \map \sgn {1, 2, 3} a_{1 1} a_{2 2} a_{3 3}\)
    \(\ds \) \(\) \(\, \ds + \, \) \(\ds \map \sgn {1, 3, 2} a_{1 1} a_{2 3} a_{3 2}\)
    \(\ds \) \(\) \(\, \ds + \, \) \(\ds \map \sgn {2, 1, 3} a_{1 2} a_{2 1} a_{3 3}\)
    \(\ds \) \(\) \(\, \ds + \, \) \(\ds \map \sgn {2, 3, 1} a_{1 2} a_{2 3} a_{3 1}\)
    \(\ds \) \(\) \(\, \ds + \, \) \(\ds \map \sgn {3, 1, 2} a_{1 3} a_{2 1} a_{3 2}\)
    \(\ds \) \(\) \(\, \ds + \, \) \(\ds \map \sgn {3, 2, 1} a_{1 3} a_{2 2} a_{3 1}\)
    \(\ds \) \(=\) \(\ds a_{1 1} a_{2 2} a_{3 3}\)
    \(\ds \) \(\) \(\, \ds - \, \) \(\ds a_{1 1} a_{2 3} a_{3 2}\)
    \(\ds \) \(\) \(\, \ds - \, \) \(\ds a_{1 2} a_{2 1} a_{3 3}\)
    \(\ds \) \(\) \(\, \ds + \, \) \(\ds a_{1 2} a_{2 3} a_{3 1}\)
    \(\ds \) \(\) \(\, \ds + \, \) \(\ds a_{1 3} a_{2 1} a_{3 2}\)
    \(\ds \) \(\) \(\, \ds - \, \) \(\ds a_{1 3} a_{2 2} a_{3 1}\)

    and thence in a single expression as:

    $\ds \map \det {\mathbf A} = \frac 1 6 \sum_{i \mathop = 1}^3 \sum_{j \mathop = 1}^3 \sum_{k \mathop = 1}^3 \sum_{r \mathop = 1}^3 \sum_{s \mathop = 1}^3 \sum_{t \mathop = 1}^3 \map \sgn {i, j, k} \map \sgn {r, s, t} a_{i r} a_{j s} a_{k t}$

    where $\map \sgn {i, j, k}$ is the sign of the permutation $\tuple {i, j, k}$ of the set $\set {1, 2, 3}$.


    The values of the various instances of $\map \sgn {\lambda_1, \lambda_2, \lambda_3}$ are obtained by applications of Parity of K-Cycle.


    Also see


    Sources

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