“determinant”的概念、定义、翻译、参考文献-科学参考

determinant

Mathematics

For the square matrix A, the determinant of A, denoted by detA or |A|, can be defined as follows.

The determinant of the 1×1 matrix [a] is simply equal to a. If A is the 2×2 matrix below, then detA = ad−bc, and the determinant can also be written as shown:

$A = [\begin{matrix} a & b \\ c & d \end{matrix}], det A = | \begin{matrix} a & b \\ c & d \end{matrix} | .$

If A is a 3×3 matrix [a_ij], then det A, which may be denoted by

$| \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} |,$

is given by

$det A = a_{11} | \begin{matrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{matrix} | - a_{12} | \begin{matrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{matrix} | + a_{13} | \begin{matrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{matrix} | .$

Notice how each 2×2 determinant occurring here is obtained by deleting the row and column containing the entry by which the 2×2 determinant is multiplied. This expression for the determinant of a 3×3 matrix can be written a₁₁A₁₁ + a₁₂A₁₂ + a₁₃A₁₃, where A_ij is the cofactor of a_ij. This is the evaluation of detA, ‘by the first row’. In fact, detA may be found by using evaluation by any row or column: for example, a₃₁A₃₁ + a₃₂A₃₂ + a₃₃A₃₃ is the evaluation by the third row, and a₁₂A₁₂ + a₂₂A₂₂ + a₃₂A₃₂ is the evaluation by the second column. The determinant of an n×n matrix A may be defined similarly, as a₁₁A₁₁ + a₁₂A₁₂ + … + a_1nA_1n, and the same value is obtained using a similar evaluation by any row or column. These are known as the Laplace expansions of the determinant. However, calculating determinants this way is laborious; using elementary operations is much more efficient. The following properties hold:
1. (i) If two rows (or two columns) of a square matrix A are identical, then detA = 0.
2. (ii) If two rows (or two columns) of a square matrix A are interchanged, then only the sign of detA is changed.
3. (iii) The value of detA is unchanged if a multiple of one row is added to another row, or if a multiple of one column is added to another column.
4. (iv) If A and B are square matrices of the same order, then det(AB)=(detA) (detB).
5. (v) If A is invertible, then det(A⁻¹)=(detA)⁻¹.
6. (vi) If A is an n×n matrix, then det kA = kⁿdetA.
7. (vii) detA^T = detA, where A^T is the transpose of A.
8. (viii) For an n×n matrix A, the map x ↦Ax scales area/volume by |detA| and is sense-preserving if detA > 0 and sense-reversing if detA < 0.

Statistics

See matrix.

Computer

A number associated with a square matrix of numbers. The determinant of an n×n matrix A is denoted by det(A) or ‖A‖ and given by
$\underset{σ}{Σ} par (σ) a_{1} σ_{1} a_{2} σ_{2} \dots a_{n} σ_{n}$
where the sum is taken over all n! permutations
$σ = σ_{1} σ_{2} \dots σ_{n}$
of the integers 1,2,…,n. par(σ‎), the parity of Σ‎, is either +1 or −1 depending on whether Σ‎ is an even permutation or an odd permutation.

Electronics and Electrical Engineering

A number that can be calculated for any square matrix, and which provides important information about the nature of that matrix, or any system it is being used to model. For a 2 × 2 matrix of the form
$M = | \begin{matrix} a & b \\ c & d \end{matrix} |$
the determinant is given by $d e t (M) = a d - b c$

单词	determinant
释义	determinant Mathematics For the square matrix A, the determinant of A, denoted by detA or \|A\|, can be defined as follows. The determinant of the 1×1 matrix [a] is simply equal to a. If A is the 2×2 matrix below, then detA = ad−bc, and the determinant can also be written as shown: $A = [\begin{matrix} a & b \\ c & d \end{matrix}], det A = \| \begin{matrix} a & b \\ c & d \end{matrix} \| .$ If A is a 3×3 matrix [a_ij], then det A, which may be denoted by $\| \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} \|,$ is given by $det A = a_{11} \| \begin{matrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{matrix} \| - a_{12} \| \begin{matrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{matrix} \| + a_{13} \| \begin{matrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{matrix} \| .$ Notice how each 2×2 determinant occurring here is obtained by deleting the row and column containing the entry by which the 2×2 determinant is multiplied. This expression for the determinant of a 3×3 matrix can be written a₁₁A₁₁ + a₁₂A₁₂ + a₁₃A₁₃, where A_ij is the cofactor of a_ij. This is the evaluation of detA, ‘by the first row’. In fact, detA may be found by using evaluation by any row or column: for example, a₃₁A₃₁ + a₃₂A₃₂ + a₃₃A₃₃ is the evaluation by the third row, and a₁₂A₁₂ + a₂₂A₂₂ + a₃₂A₃₂ is the evaluation by the second column. The determinant of an n×n matrix A may be defined similarly, as a₁₁A₁₁ + a₁₂A₁₂ + … + a_1nA_1n, and the same value is obtained using a similar evaluation by any row or column. These are known as the Laplace expansions of the determinant. However, calculating determinants this way is laborious; using elementary operations is much more efficient. The following properties hold: (i) If two rows (or two columns) of a square matrix A are identical, then detA = 0. (ii) If two rows (or two columns) of a square matrix A are interchanged, then only the sign of detA is changed. (iii) The value of detA is unchanged if a multiple of one row is added to another row, or if a multiple of one column is added to another column. (iv) If A and B are square matrices of the same order, then det(AB)=(detA) (detB). (v) If A is invertible, then det(A⁻¹)=(detA)⁻¹. (vi) If A is an n×n matrix, then det kA = kⁿdetA. (vii) detA^T = detA, where A^T is the transpose of A. (viii) For an n×n matrix A, the map x ↦Ax scales area/volume by \|detA\| and is sense-preserving if detA > 0 and sense-reversing if detA < 0. Statistics See matrix. Computer A number associated with a square matrix of numbers. The determinant of an n×n matrix A is denoted by det(A) or ‖A‖ and given by $\underset{σ}{Σ} par (σ) a_{1} σ_{1} a_{2} σ_{2} \dots a_{n} σ_{n}$ where the sum is taken over all n! permutations $σ = σ_{1} σ_{2} \dots σ_{n}$ of the integers 1,2,…,n. par(σ‎), the parity of Σ‎, is either +1 or −1 depending on whether Σ‎ is an even permutation or an odd permutation. Electronics and Electrical Engineering A number that can be calculated for any square matrix, and which provides important information about the nature of that matrix, or any system it is being used to model. For a 2 × 2 matrix of the form $M = \| \begin{matrix} a & b \\ c & d \end{matrix} \|$ the determinant is given by $d e t (M) = a d - b c$
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