Matrix

What is a matrix?

A rectangular array of numbers, symbols, or expressions arranged in rows and columns.

What is the dimension of a matrix?

Defined as $m \times n$ where $m$ is the number of rows and $n$ is the number of columns.

What is a square matrix?

A matrix where the number of rows equals the number of columns (i.e., $n \times n$).

What is a Transpose matrix?

Created by swapping the rows and columns of the original matrix; denoted as $A^T$.

What is the requirement for matrix addition?

Both matrices must have identical dimensions (e.g., $m \times n$).

How do you calculate the matrix product?

Multiply corresponding elements of a row from the first matrix by a column from the second and sum them up.

What is the result dimension of multiplying $A (m \times n)$ and $B (n \times p)$?

The result is a matrix of dimension $m \times p$.

What is an Identity matrix?

A square matrix with 1s on the main diagonal and 0s elsewhere; denoted as $I_n$.

What is the logical meaning of the Identity matrix in multiplication?

It acts like the number 1, meaning $A \times I = A$ for any matrix A.

What defines a Symmetric matrix?

A square matrix that is equal to its transpose, i.e., $A = A^T$.

What is a Join in Zero-One Matrices?

A Boolean 'OR' operation performed element-wise, resulting in 1 if at least one corresponding element is 1.

What is the condition for the Meet operation in Zero-One Matrices?

It results in 1 only if both matrices have 1 at the same position.

What does the Boolean Product of two matrices entail?

A product calculated using the AND operation for multiplication and OR for addition.

Can you find the power of a non-square matrix like $A (2 \times 3)$?

No, powers are only defined for square matrices (i.e., $n \times n$).

What is the proof logic for the Identity property?

To prove $AI = A$, show that $\sum a_{ik} \delta_{kj} = a_{ij}$ where $\delta$ is the Kronecker delta.

What is the Transpose of a Product rule?

$(AB)^T = B^T A^T$, reversing the order of multiplication.