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Linear Algebra

Matrices

An \(m\times n\) matrix A is a rectangular array with \(m\) and \(m\) columns. We denote the entry in the i-th row and j-th column by \(a_{ij}\). That is,

\[\left[\begin{array}{cccccc} a_{11} & a_{12} & \ldots & a_{1j} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2j} & \dots & a_{2n} \\ \vdots & \vdots && \vdots && \vdots \\ a_{i1} & a_{i2} & \ldots & a_{ij} & \ldots & a_{in} \\ \vdots & \vdots && \vdots && \vdots \\ a_{m1} & a_{m2} & \ldots & a_{mj} & \ldots & a_{mn} \end{array}\right]\]

Two \(m\times n\) matrices \(A\) and \(B\) are equal if \(a_{ij} =b_{ij}\) for all \(1\leq i\leq m\), \(1 \leq j\leq n\).

The set of all \(m\times n\) matrices with real entries is denoted by \(M_{m\times n} (\mathbb{R})\).

A matrix in \(M_{n\times n}(\mathbb{R})\) (the number of rows of the matrix is equal to the number of columns) is called a square matrix.

The following is a \(3\times 3\) square matrix.

\begin{bmatrix} a &b&c\\d&e&f\\g&h&i\end{bmatrix}

A matrix \(U \in M_{n\times n}(\mathbb{R})\) is said to be upper triangular if \(u_{ij}=0\) whenever \(i>j\).

A matrix \(L \in M_{n\times n}(\mathbb{R})\) is said to be lower triangular if \(l_{ij}=0\) whenever \(i<j\).

Upper triangle matrices

\[\begin{bmatrix} 3&4\\0&1\end{bmatrix} \qquad \begin{bmatrix} 5&2&1\\0&2&3\\0&0&1\end{bmatrix} \]

 

Lower triangle matrices

\[\begin{bmatrix} -3&0\\4&1\end{bmatrix} \qquad \begin{bmatrix} 0&0&0\\1&2&0\\0&5&0\end{bmatrix} \]

Upper and Lower triangle matrix

\[ \begin{bmatrix} 3&0&0\\0&2&0\\0&0&1\end{bmatrix} \]

Operations on Matrices

Let \(A, B\in M_{n\times n} (mathbb{R})\). We define addition of matrices by 

\[(A+B)_{ij} = (A)_{ij} + (B)_{ij} \]

We define scalar multiplication of \(A\) by a scalar \(t\in\mathbb{R}\) by

\[(tA)_{ij}=t(A)_{ij}\]

By the definition above, matrix addition can only be performed on matrices of the same size. See an example in the video.

Example: Multiply matrix \(A\) by the scalar multiple \(k=-3\)

\[A=\begin{bmatrix} 3 & 0 &2\\-2&0&0\\0&-3&4\end{bmatrix}\]

\[3A=\begin{bmatrix} -9 & 0 &-6\\6&0&0\\0&9&-12\end{bmatrix}\]

Transpose

Columns and rows can be switched by finding the transpose of a matrix.

Let \(A\in = M_{m\times n}\in(\mathbb{R})\). The transpose of \(A\), is the \(n\times m\) matrix, denoted \(A^T\), whose ij-th entry is the ji-th entry of \(A\). That is \[(A^T)_{ij} = (A)_{ji}\]

Example: Find the transpose of \(A\)

\[A=\begin{bmatrix} -3&6&2\\0&-2&1\end{bmatrix}\]

Solution

\[A^T=\begin{bmatrix} -3&0\\6&-2\\2&1\end{bmatrix}\]

Properties of Transpose Matrices

If \(A\) and \(B\) are matrices, and \(s\in\mathbb{R}\), then

  1. \((A^T)^T=A\)
  2. \((A+B)^T=A^T+B^T\)
  3. \((sA)^T=sA^T\)

Matrix Multiplication

Let \(A\in M_{m\times n}(\mathbb{R})\) with rows \(\vec{a}_1,\ldots,\vec{a}_m\) and let \(B\in M_{n\times p}(\mathbb{R})\) with columns \(\vec{b}_1,\ldots,\vec{b}_p\) . We define \(AB\) to be the \(m\times p\) matrix whose ij-th entry is 

\[(AB)_{ij} = \vec{a}_i \cdot \vec{b}_{j}\]

There are key elements to notice during matrix multiplication:

  • Order matters in matrix multiplication (e.g., \(AB \neq BA\))
  • The product \(AB\) of two matrices \(A\) and \(B\), can only be performed if the number of columns in \(A\) is the same as the number of rows in \(B\).
  • The product \(AB\) has the same number of rows as matrix \(A\) and the same number of columns as matrix \(B\).
  • The element \(a_{ij}\) is found by multiplying row \(i\) in matrix \(A\) with column \(j\) in matrix \(B\).

Identity Matrix

When performing an operation with an identity, the resultant is the same value. For example, adding 0 to any number gives you the same number. Multiplying any number by 1 also gives you the same number. How do these identities look like with matrices? The addition identity is a matrix with 0 in all its elements. However, it is not true that having a matrix with all 1s as its elements will create a multiplication identity. Instead, the identity matrix for multiplication has 1s only in the diagonal and 0s everywhere else.

The identity matrix for an \(n\times n\) matrix is \[I=\begin{bmatrix}1&\ldots&\ldots&0\\0&1&\ldots&0\\ \vdots&&\ddots&\vdots\\0&\ldots&\ldots&1\end{bmatrix}\]

For example, the identity matrix \(I\) in a \(3\times 3\) matrix is

\begin{bmatrix} 1&0&0\\0&1&0\\0&0&1 \end{bmatrix}

The identity matrix \(I\) in a \(4\times 4\) matrix is

\begin{bmatrix} 1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1 \end{bmatrix}

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Designed by Matthew Cheung. This work is licensed under a Creative Commons Attribution 4.0 International License.
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