线性代数:行列式性质

The contents are from chapter 5 section 1 of Introduction to Linear Algebra, Book of Gilbert Strang, edition 4th.

• The properties of determinants

1.Key ideas

1. The determinant is defined by \(\det I = 1\), sign reversal, and linearity in each row.
2. After elimination \(\det A\) is \(\pm\)(product of the pivots).
3. The determinant is zero exactly when A is not invertible.
4. Two remarkable properties are \(\det{AB}=(\det A)(\det B)\) and \(\det{A^T}=\det A\).

2.Introduction

The determinant of an n by n matrix can be found in three ways:

• pivot formula: multiply the n pivots(times 1 or -1)
• “big” formula: add up n! terms(times 1 or -1)
• cofactor formula: combine n smaller determinants(times 1 or -1)

Let me show a simple example to get the determinant of matrix A.

The determinant changes sign when tow rows (or two columns) are exchanged.

Row and column are equal.

Applications：

1. Determinants give \(A^{-1}\) and \(A^{-1}b\)(this formula is called Cramer’s Rule).
2. When the edges of a box are the rows of A, the volume is \(\mid \det A \mid\).
3. For n special numbers \(\lambda\), called eigenvalues, the determinants of \(A - \lambda I\) is zero. This is a truly important application and it fills Chapter 6.

3.The properties of the determinant

1. \[\det I = 1\]
2. The determinant changes sign when two rows are changed(sign reversal).交换行，行列式值符号改变。（列也是一样）
3. Key：The determinant is a linear function of each row separately.
   • a. \(\begin{vmatrix} ta & tb \\ c & d \end{vmatrix} = t \begin{vmatrix} a & b \\ c & d\end{vmatrix}\)
   • b. \(\begin{vmatrix} a+a' & b+b' \\ c & d\end{vmatrix} = \begin{vmatrix} a & b \\ c & d\end{vmatrix} + \begin{vmatrix} a' & b' \\ c & d\end{vmatrix}\)
4. If two rows of A are equal, then \(\det A = 0\).(两行相等, 行列式为0)
5. Key: Subtracting a multiple of one row from another row leaves \(\det A\) unchanged.
   • Using rule 3 + 4 to prove this.
   • Example: \(\begin{vmatrix} a & b \\ c-la & d-lb \end{vmatrix} = \begin{vmatrix} a & b \\ c & d\end{vmatrix}\).
6. A matrix with a row of zeros has \(\det A = 0\) 一行全为0, 则行列式为0.
7. If A is triangular then \(\det A = a_{11}a_{22} \cdots a_{nn} =\) product of diagonal entries.
   • ex: \(\begin{vmatrix} a & b \\ 0 & d \end{vmatrix} = ad, \begin{vmatrix} a & 0 \\ c & d \end{vmatrix} = ad\).
8. If A is singular then \(\det A = 0\). If A is invertible then \(\det A \neq 0\).
9. \(\det{AB} = \det A \det B\).
   • ex: \(\begin{vmatrix} a & b \\ c & d \end{vmatrix} \begin{vmatrix} p & q \\ r & s \end{vmatrix} = \begin{vmatrix} ap+br & aq+bs \\ cp+dr & cq+ds \end{vmatrix}\).
10. \(\det{A^T} = \det A\).
    • Prove:
    • 分解:\(A=LU\), 其中\(L\)为下三角矩阵, \(U\)为上三角矩阵.
    • \[A^T=U^T L^T\]
    • \(\mid U^T \mid \mid L \mid = \mid L \mid \mid U \mid\).

注: 1-8 为行列式自身的性质, 知道这些可以推导性质.

@Anifacc
2017-06-03

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