Determinants

Andrew Thangaraj

Aug-Nov 2020

Recap

  • Vector space \(V\) over a scalar field \(F\)
    • \(F\): real field \(\mathbb{R}\) or complex field \(\mathbb{C}\) in this course
  • \(m\times n\) matrix A represents a linear map \(T:F^n\to F^m\)
    • null\((A)=\) null \(T=\{v\in V:Tv=0\}\), colspace\((A)\) = range \(T=\{Tv:v\in V\}\)
    • dim null \(T\) + dim range \(T\) = dim \(V\)
  • Linear equation: \(Ax=b\)
    • Solved using elementary row operations
    • Solution (if it exists): \(u+\) null\((A)\)
  • Linear map \(T\) induces a one-to-one map \(V/\text{null }T\to \text{range }T\)
  • Four fundamental subspaces of a matrix
    • Column space, row space, null space, left null space

Determinants

\(A:n\times n\) matrix written as \(A=[v_1;\ldots;v_n]\), where \(v_j\) is the \(j\)-th row of \(A\).

\(v_j\): row vector of length \(n\)

det: \(F^{n,n}\to F\) is a function from square matrices to the field \(F\) satisfying the following conditions or defining properties:

  1. Identity: det\((I)=1\)

  2. Row scaling: det\(([v_1;\ldots;cv_j;\ldots,v_n])=c\) det\(([v_1;\ldots;v_j;\ldots;v_n])\)

  3. Row linearity: det\(([v_1;\ldots;v_j+v'_j;\ldots;v_n])=\) det\(([v_1;\ldots;v_j;\ldots;v_n])+\) det\(([v_1;\ldots;v'_j;\ldots;v_n])\)

  4. Equal rows: det\(([v_1;\ldots;v_n])=0\), if any two rows are equal

  • why such an intricate definition?

  • is there such a function?

Geometric connection

\(A=\begin{bmatrix} a&b\\ c&d \end{bmatrix}\), det\((A)=ad-bc\)

  • Area of parallelogram formed by \((a,b),(c,d)\) \(=\) \(|\)det\((A)|\)
  • Defining properties satisfied by area function

\(A=\begin{bmatrix} a&b&c\\ d&e&f\\ g&h&i \end{bmatrix}\), det\((A)=aei+bfg+cdh-gec-hfa-idb\)

  • Volume of cuboid formed by \((a,b,c),(d,e,f),(g,h,i)\) \(=\) \(|\)det\((A)|\)
  • Defining properties satisfied by volume function

\(A:n\times n\) matrix, det\((A)=?\)

  • Volume of \(n\)-dimensional parallelepiped formed by rows
  • Defining properties satisfied

Further properties of determinants

Zero row: det\(([v_1;\ldots;v_n])=0\), if any of the rows is equal to all zeros

Proof

  • Use row scaling property with \(c=-1\) on all-zero row

Row operation: det\(([v_1+cv_j;\ldots;v_j;\ldots;v_n])=\text{det}([v_1;\ldots;v_j;\ldots;v_n])\)

Proof

  • Use row addition, scaling and equal row properties

Dependent rows: det\(([v_1;\ldots;v_n])=0\), if the rows or columns are linearly dependent

Proof

  • Replace \(v_j\) by \(v_j+a_1v_1+\cdots+a_{j-1}v_{j-1}=0\)
  • Take determinant and use defining properties

Row swap: If two rows are interchanged, determinant gets multiplied by \(-1\)

Proof

  • Consider det\(([v_1+v_2;v_1+v_2;\ldots])\)

Determinants and elementary row operations

Elementary row operators: Let \(E\) be an elementary row operator \[\text{det}(E)=\begin{cases} c&\text{ if row scaling by }c\\ -1&\text{ if row swap}\\ 1&\text{ if row }i = \text{row }i + c(\text{row }j) \end{cases}\]

Product of elementary row operators and a matrix \[\text{det}(EA)=\text{det}(E)\text{det}(A)\]

\[\text{det }\left(\left(\prod_i E_i\right) A\right) = \left(\prod_i \text{det}(E_i)\right)\text{det}(A)\]

  • Proof of last result by induction on \(i\)

Compute determinants using properties

Diagonal matrix: det \(\begin{bmatrix}d_1&0&\cdots&0\\ 0&d_2&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&d_n\end{bmatrix}=d_1d_2\cdots d_n\)

Non-invertible matrix: det \(=0\)

Invertible matrix: Row reduce to identity

There exist elementary row operators \(E_i\) such that \[\left(\prod_i E_i\right) A = I\]

  • Take determinants and use elementary row operator property

det\((A)=(-1)^{n_s}\dfrac{1}{\left(\prod_j c_j\right)}\)

Determinant of product of two matrices

\(A,B\): two square matrices

det\((AB)=\) det\((A)\) det\((B)\)

Proof

  • \(A\) or \(B\) non-invertible

    • \(AB\) is also non-invertible

    • det\((AB)=0=\) det\((A)\) det\((B)\)

  • \(A\) and \(B\) invertible

    • \(\left(\prod_i E_i\right) A = I\), \(\left(\prod_j F_j\right) B = I\)

    • \(\left(\prod_j F_j\right)\left(\prod_i E_i\right) AB = I\)

Corollary: If \(A\) is invertible, det\((A^{-1})=\dfrac{1}{\text{det}(A)}\)

Elementary column operations and transpose

Elementary column operators: \(E^T\), where \(E\) is an elementary row operator

det\((E^T)=\) det\((E)\)

det\((A^T)=\text{det}(A)\)

Proof

  • \(A\): non-invertible

    • \(A^T\) is non-invertible

    • det\((A^T)=0=\text{det}(A)\)

  • \(A\): invertible

    • \(E_1\cdots E_L A = I\) implies \(I=A^T E^T_L\cdots E^T_1\)

    • Take determinants

Determinant function definition

det: \(F^{n,n}\to F\) is a function from square matrices to the field \(F\) satisfying the following conditions or defining properties:

  1. Identity: det\((I)=1\)

  2. Row scaling: det\(([v_1;\ldots;cv_j;\ldots,v_n])=c\) det\(([v_1;\ldots;v_j;\ldots;v_n])\)

  3. Row linearity: det\(([v_1;\ldots;v_j+v'_j;\ldots;v_n])=\) det\(([v_1;\ldots;v_j;\ldots;v_n])+\) det\(([v_1;\ldots;v'_j;\ldots;v_n])\)

  4. Equal rows: det\(([v_1;\ldots;v_n])=0\), if any two rows are equal

  • Unique function that satisfies all of the above properties

  • Co-factor expansion along any row or column

  • Permutation formula