4.2 Matrix Multiplication and Properties

Matrix multiplication is a crucial operation in linear algebra, allowing us to combine matrices and transform data. It's not just about multiplying numbers—it's a powerful tool for solving complex problems in various fields, from computer graphics to economics.

Understanding matrix multiplication is key to grasping more advanced concepts in linear algebra. It's the foundation for many applications, including linear transformations, Markov chains, and cryptography. Mastering this operation opens doors to a wide range of mathematical and practical applications.

Matrix Multiplication

Definition and Notation

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• Matrix multiplication is a binary operation that produces a matrix from two matrices
• Denoted by the symbol "·" or by writing the two matrices next to each other with no symbol in between
• The resulting matrix, known as the matrix product, has the same number of rows as the first matrix and the same number of columns as the second matrix

Computation Process

• Each entry in the matrix product is calculated by multiplying corresponding entries from one row of the first matrix and one column of the second matrix and then adding the results
• Formally, if C = AB, then each entry $c_{ij}$ in the product matrix C is calculated as follows: $c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$, where $i = 1, 2, ..., m$ and $j = 1, 2, ..., p$
• To compute the product efficiently, use the "row by column" method: multiply each row of the first matrix by each column of the second matrix, and sum the results to obtain each entry in the product matrix

Matrix Compatibility for Multiplication

Size Requirements

• For matrix multiplication to be possible, the matrices must be compatible in size
• The number of columns in the first matrix must equal the number of rows in the second matrix
• If A is an $m \times n$ matrix and B is an $n \times p$ matrix, then their product AB is an $m \times p$ matrix

Undefined Products

• If the number of columns in the first matrix does not equal the number of rows in the second matrix, then the matrices are not compatible for multiplication, and the product is undefined
• Attempting to multiply incompatible matrices will result in an error or undefined operation

Matrix Multiplication Computation

Resulting Matrix Dimensions

• To compute the product of two matrices A and B, where A is an $m \times n$ matrix and B is an $n \times p$ matrix, the resulting matrix C will be an $m \times p$ matrix
• The number of rows in the product matrix is determined by the number of rows in the first matrix, while the number of columns is determined by the number of columns in the second matrix

Entry Calculation

• Each entry $c_{ij}$ in the product matrix C is calculated by multiplying the entries from the $i$-th row of matrix A with the corresponding entries from the $j$-th column of matrix B and then adding the results
• The formal calculation for each entry is given by $c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$, where $i = 1, 2, ..., m$ and $j = 1, 2, ..., p$
• This process is repeated for each entry in the product matrix until all entries have been calculated

Row by Column Method

• To compute the product efficiently, use the "row by column" method
• Multiply each row of the first matrix by each column of the second matrix, and sum the results to obtain each entry in the product matrix
• This method avoids unnecessary calculations and provides a systematic approach to computing the matrix product

Properties of Matrix Multiplication

Non-Commutativity

• Matrix multiplication is not commutative: AB ≠ BA in general
• The order of the matrices in multiplication matters and cannot be interchanged without potentially changing the result
• Example: Let $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$. Then, $AB = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}$ while $BA = \begin{pmatrix} 23 & 34 \\ 31 & 46 \end{pmatrix}$, showing that AB ≠ BA.

Associativity and Identity

• Matrix multiplication is associative: (AB)C = A(BC) for compatible matrices A, B, and C
• The identity matrix I acts as the multiplicative identity: AI = IA = A for any matrix A
• Example: Let $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, $B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$, and $C = \begin{pmatrix} 9 & 10 \\ 11 & 12 \end{pmatrix}$. Then, $(AB)C = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}\begin{pmatrix} 9 & 10 \\ 11 & 12 \end{pmatrix} = \begin{pmatrix} 411 & 454 \\ 939 & 1038 \end{pmatrix}$ and $A(BC) = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\begin{pmatrix} 95 & 106 \\ 119 & 132 \end{pmatrix} = \begin{pmatrix} 411 & 454 \\ 939 & 1038 \end{pmatrix}$, confirming that (AB)C = A(BC).

Distributivity and Transpose

• The distributive property holds for matrix multiplication: A(B + C) = AB + AC and (A + B)C = AC + BC for compatible matrices A, B, and C
• The transpose of a product is equal to the product of the transposes in reverse order: $(AB)^T = B^TA^T$
• Example: Let $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$. Then, $(AB)^T = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}^T = \begin{pmatrix} 19 & 43 \\ 22 & 50 \end{pmatrix}$ and $B^TA^T = \begin{pmatrix} 5 & 7 \\ 6 & 8 \end{pmatrix}\begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix} = \begin{pmatrix} 19 & 43 \\ 22 & 50 \end{pmatrix}$, confirming that $(AB)^T = B^TA^T$.

Inverse of a Product

• The inverse of a product is equal to the product of the inverses in reverse order: $(AB)^{-1} = B^{-1}A^{-1}$, provided A and B are invertible matrices
• This property allows for the simplification of matrix equations and the solution of systems of linear equations
• Example: Let $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix}$. Then, $AB = \begin{pmatrix} -2 & 0 \\ 0 & -2 \end{pmatrix}$, and $(AB)^{-1} = \begin{pmatrix} -\frac{1}{2} & 0 \\ 0 & -\frac{1}{2} \end{pmatrix}$. Also, $A^{-1} = \begin{pmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{pmatrix}$ and $B^{-1} = \begin{pmatrix} \frac{1}{2} & 1 \\ \frac{3}{2} & 2 \end{pmatrix}$, so $B^{-1}A^{-1} = \begin{pmatrix} \frac{1}{2} & 1 \\ \frac{3}{2} & 2 \end{pmatrix}\begin{pmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{pmatrix} = \begin{pmatrix} -\frac{1}{2} & 0 \\ 0 & -\frac{1}{2} \end{pmatrix}$, confirming that $(AB)^{-1} = B^{-1}A^{-1}$.

Applications of Matrix Multiplication

Linear Transformations

• Apply matrix multiplication to solve problems in various fields, such as physics, computer graphics, and economics
• Use matrix multiplication to calculate the composition of linear transformations, where each matrix represents a transformation
• Example: In computer graphics, matrices can represent translations, rotations, and scaling operations. Multiplying these matrices together allows for the combination of multiple transformations into a single matrix operation.

Markov Chains

• Employ matrix multiplication in the context of Markov chains to model and analyze systems that transition between different states
• The transition matrix of a Markov chain can be multiplied by a state vector to determine the probability distribution of the system after a given number of steps
• Example: In a weather model, the states could represent "Sunny," "Cloudy," and "Rainy," and the transition matrix would contain the probabilities of moving from one state to another. Multiplying the transition matrix by the current state vector would give the probability distribution of the weather for the next day.

Graphs and Networks

• Utilize matrix multiplication in the study of graphs and networks, where matrices can represent adjacency relations or weights between nodes
• Multiplying the adjacency matrix of a graph by itself can reveal information about the connections between nodes, such as the number of paths of a certain length between two nodes
• Example: In a social network, the adjacency matrix could represent connections between users. Multiplying the adjacency matrix by itself would give the number of mutual friends between each pair of users.

Cryptography

• Apply matrix multiplication in the field of cryptography for encrypting and decrypting messages using matrix-based algorithms
• The Hill cipher is an example of a cryptographic algorithm that uses matrix multiplication to encrypt and decrypt text messages
• Example: In the Hill cipher, each letter of the alphabet is assigned a numeric value, and the plaintext is divided into blocks of a fixed size. Each block is then multiplied by a secret key matrix to produce the ciphertext. To decrypt, the ciphertext is multiplied by the inverse of the key matrix.