9.10 Polynomials and Matrices Tutorial

Matrices

Let \(\mathbb{R}\) be a ring \((\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C})\).

Definition: A matrix \(A\) with size \(m \times n\) with coefficients in \(\mathbb{R}\) is a rectangular array of elements in \(\mathbb{R}\) of the form

\[ A = \begin{pmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{pmatrix} \]

where \(m\) is the number of rows and \(n\) is the number of columns.

The matrix \(A\) is denoted \(A = (a_{ij})_{m \times n}\) and the set of all matrices of size \(m \times n\) with coefficients in \(\mathbb{R}\) is denoted \(M_{m \times n}(\mathbb{R})\).

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Definition: The identity matrix of size \(n\), denoted \(I_n\), is defined by \(I_n = (\delta_{ij})_{n \times n}\), where

\[ \delta_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases} \]

Examples:

\[ I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad I_4 = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

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Definition:

Definition: The main diagonal of a square matrix \(A = (a_{ij})_{n \times n}\) consists of the elements \(a_{11}, a_{22}, \dots, a_{nn}\).

Definition: A diagonal matrix is a square matrix \(A = (a_{ij})_{n \times n}\) where \(a_{ij} = 0\) for \(i \neq j\), that is, a square matrix such that the elements outside the main diagonal are \(0\).

Examples:

\[ A = \begin{pmatrix} 8 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 7 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -9 \end{pmatrix} \]

Definition: A scalar matrix is a diagonal matrix in which the elements of the main diagonal are equal.

Examples:

\[ A = \begin{pmatrix} -2 & 0 \\ 0 & -2 \end{pmatrix}, \quad B = \begin{pmatrix} 7 & 0 & 0 \\ 0 & 7 & 0 \\ 0 & 0 & 7 \end{pmatrix} \]

Definition: Given a matrix \(A = (a_{ij})_{m \times n}\) and \(\alpha \in \mathbb{R}\), the scalar product of \(A\) by \(\alpha\) is the matrix \(\alpha A = (b_{ij})_{m \times n}\) defined by:

\[ b_{ij} = \alpha a_{ij} \quad \text{for all} \ 1 \leq i \leq m, \ 1 \leq j \leq n \]

Example: If \(A = \begin{pmatrix} 1 & 7 \\ 2 & 6 \\ 3 & 5 \end{pmatrix}\), then \(2A = \begin{pmatrix} 2 & 14 \\ 4 & 12 \\ 6 & 10 \end{pmatrix}\)

Theorem: Let \(A\), \(B\), \(C \in M_{m \times n}(\mathbb{R})\) and let \(\alpha, \beta \in \mathbb{R}\), then:

1. \((A + B) + C = A + (B + C)\)

2. \(A + B = B + A\)

3. \(A + 0 = A\)

4. \(A + (-A) = 0\)

5. \(\alpha(A + B) = \alpha A + \alpha B\)

6. \((\alpha + \beta)A = \alpha A + \beta A\)

7. \((\alpha \beta) A = \alpha (\beta A)\)

8. \(1 A = A\)

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Definition: Given two matrices \(A = (a_{ij})_{m \times n}\) and \(B = (b_{jk})_{n \times p}\), the product of \(A\) and \(B\) is the matrix \(AB = (c_{ik})_{m \times p}\) defined by:

\[ c_{ik} = \sum_{j=1}^{n} a_{ij}b_{jk} \]

for all \(1 \leq i \leq m\), \(1 \leq k \leq p\).

Example: Let \(A = \begin{pmatrix} 1 & 2 & -1 \\ 3 & 1 & 4 \end{pmatrix}\) and \(B = \begin{pmatrix} -2 & 5 \\ 4 & -3 \\ 2 & 1 \end{pmatrix}\), then

\[ AB = \begin{pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{pmatrix} \]

Where:

\[ c_{11} = \sum_{j=1}^{3} a_{ij} b_{j1} = a_{11} b_{11} + a_{12} b_{21} + a_{13} b_{31} = (1)(-2) + (2)(4) + (-1)(2) = 4 \]

\[ c_{12} = \sum_{j=1}^{3} a_{ij} b_{j2} = a_{11} b_{12} + a_{12} b_{22} + a_{13} b_{32} = (1)(5) + (2)(-3) + (-1)(1) = -2 \]

\[ c_{21} = \sum_{j=1}^{3} a_{2j} b_{j1} = a_{21} b_{11} + a_{22} b_{21} + a_{23} b_{31} = (3)(-2) + (1)(4) + (4)(2) = 6 \]

\[ c_{22} = \sum_{j=1}^{3} a_{2j} b_{j2} = a_{21} b_{12} + a_{22} b_{22} + a_{23} b_{32} = (3)(5) + (1)(-3) + (4)(1) = 16 \]

Theorem.
Let \(A\), \(B\), \(C\) be matrices with sizes such that the following equations are valid, then:

(1) \((AB)C = A(BC)\)

(2) \((A + B)C = AC + BC\)

(3) \(A(B + C) = AB + AC\)

(4) \(\alpha(AB) = (\alpha A)B = A(\alpha B)\)

(5) \(AI_n = A = I_mA\) if \(A \in M_{m \times n}(\mathbb{R})\)