Ch 2.1 - Matrix Operations

Flashcards covering key definitions and concepts from the Matrix Operations lecture, including matrix types, arithmetic, multiplication, and properties.

Matrix factorizations

Techniques used to simplify matrix computations

Diagonal entries

The entries a₁₁, a₂₂, a₃₃, … that form the main diagonal of a matrix A.

Diagonal matrix

A square n x n matrix whose non-diagonal entries are all zero.

Zero matrix (0)

An m x n matrix whose entries are all zero.

Equal matrices

Two matrices are equal if they have the same size and their corresponding entries are equal.

Sum of matrices (A + B)

For m x n matrices A and B, the m x n matrix whose columns (and entries) are the sums of the corresponding columns (and entries) in A and B, defined only when A and B are the same size.

Scalar multiple (rA)

For a scalar r and a matrix A, the matrix whose columns are r times the corresponding columns in A.

Matrix multiplication (A⋅B)

The product of an m x n matrix A and an n x p matrix B, resulting in an m x p matrix whose columns are A multiplied by each column of B ([Ab₁ Ab₂ … Ab_p]). It represents the composition of linear transformations.

Row-column rule for computing A⋅B

A method where the (i, j)-entry of AB is calculated as the sum of the products of corresponding entries from row i of A and column j of B.

Identity Matrix (I_n)

A square diagonal matrix (n x n) with ones on the main diagonal and zeros elsewhere, acting as the identity element for matrix multiplication (e.g., I_m ⋅ A = A = A ⋅ I_n).

Commute (matrices)

Two matrices A and B that produce the same product regardless of order, i.e., AB = BA.

Powers of a Matrix (A^k)

For an n x n matrix A and a positive integer k, A^k denotes the product of k copies of A.

A⁰ (Matrix)

Interpreted as the identity matrix.

Transpose of a Matrix (A^T)

For an m x n matrix A, it is the n x m matrix whose columns are formed from the corresponding rows of A.