Properties of Matrices (Chapter 1) - CENUMES 313

Flashcards covering key concepts from Chapter 1: Properties of Matrices, including definitions of matrix types, transpose, and basic matrix algebra.

What is a matrix?

A rectangular array of elements arranged in rows and columns, usually denoted by a boldface capital letter (e.g., A).

How is matrix size denoted?

As n × m, meaning n rows and m columns.

What do the subscripts a_ij denote?

The element in row i and column j of a matrix.

What is a square matrix?

A matrix with the same number of rows and columns (n = m).

What is the major diagonal?

The left-to-right downward-sloping line of elements from a11 to ann.

What is a diagonal matrix?

A square matrix with all elements zero except those on the major diagonal.

What is the identity matrix?

A diagonal matrix with ones on the main diagonal; denoted I.

What is a scalar matrix?

A square matrix where all diagonal entries are equal; the identity is a special case.

What is a symmetric matrix?

A square matrix satisfying aij = aji for all i, j (A = A^T).

What is a zero (null) matrix?

A matrix in which every element is zero.

What is a vector and how are vectors represented?

A special type of matrix with either one column (column vector) or one row (row vector); bold letters are often used (e.g., X for a column vector).

What is a column vector dimension?

n × 1.

What is a row vector dimension?

1 × m.

What is a triangular matrix?

A square matrix with all elements on one side of the major diagonal equal to zero.

What is an upper triangular matrix?

All elements below the major diagonal are zero.

What is a lower triangular matrix?

All elements above the major diagonal are zero.

What is a tridiagonal matrix?

A square matrix with nonzero elements only on the main diagonal and the two diagonals surrounding it; others are zero.

What is a banded matrix?

A matrix with nonzero elements only along certain diagonals.

What is a sparse matrix?

A matrix in which most elements are zero; common in large PDE problems.

What is the transpose of a matrix A?

A^T is the matrix obtained by swapping rows and columns, with aij^T = aji (size m×n if A is n×m).

What happens to the transpose of a column vector?

It becomes a row vector, and vice versa.

What is a symmetric square matrix property?

A = A^T; equivalently aij = aji.

What is matrix addition?

Addition of corresponding elements of two matrices of equal size: A + B.

What is matrix subtraction?

Subtraction of corresponding elements: A − B.

Are matrix additions associative and commutative?

Yes: associative (A + (B + C) = (A + B) + C) and commutative (A + B = B + A).

What is matrix multiplication?

Product AB defined when A is n×m and B is m×r; C = AB with cij = sum over k of aik b_kj; result is n×r.

When are matrices conformable for multiplication?

When the number of columns of A equals the number of rows of B (A: n×m, B: m×r).

What is the size of AB if A is 4×3 and B is 3×2?

AB is a 4×2 matrix.

Are square matrices commutative under multiplication?

Not in general; AB ≠ BA.

Is matrix multiplication associative?

Yes: A(BC) = (AB)C for conformable matrices.

What is the size of F^T if F is 3×5?

5×3.

How do you obtain a transpose in practice?

Flip elements across the main diagonal; aij moves to aji; diagonal elements stay on the diagonal.

What happens to diagonal elements under transposition?

Diagonal elements a_ii remain on the diagonal after transposition.

What does conformability mean for multiplication (example concept)?

The inner dimensions must match (A: n×m, B: m×r) so that the product AB is defined.