Linear Algebra Lecture Notes Flashcards

Comprehensive practice flashcards covering vocabulary and definitions from Linear Algebra lecture notes, including matrix types, properties, rank, eigenvalues, and algebraic structures.

Matrix

A rectangular arrangement of numbers into $m$ rows and $n$ columns, where the numbers forming the matrix are called elements or entries.

Principal Diagonal

The diagonal line in a square matrix $A_{n \times n}$ formed by elements $a_{ii}$ for $1 \le i \le n$.

Super Diagonal Line

The line of elements in a square matrix located directly above the principal diagonal, denoted by elements $a_{i, i+1}$.

Sub Diagonal Line

The line of elements in a square matrix located directly below the principal diagonal, denoted by elements $a_{i, i-1}$.

Diagonal Matrix

A square matrix in which all non-diagonal elements are zero.

Scalar Matrix

A diagonal matrix in which all elements along the principal diagonal are the same.

Unit Matrix

A scalar matrix in which every element on the principal diagonal is equal to $1$.

Upper Triangular Matrix

A square matrix $[a_{ij}]_{n \times n}$ where all elements below the principal diagonal are zero, i.e., $a_{ij} = 0$ for all $i > j$.

Lower Triangular Matrix

A square matrix where all elements above the principal diagonal are zero, i.e., $a_{ij} = 0$ for all $i < j$.

Sub Matrix

A matrix obtained from a given matrix by deleting any number of rows and columns.

Total Submatrices (Distinct Entries)

For a matrix $A_{m \times n}$ with all distinct entries, the total number of submatrices is $(2^m - 1) \times (2^n - 1)$, while for identical entries, it is $m \times n$.

Consecutive Submatrix

A submatrix obtained from a given matrix by selecting only adjacent rows and adjacent columns.

Total Consecutive Submatrices

For a matrix $A_{m \times n}$ with distinct entries, the total count is calculated as $\frac{m(m+1)}{2} \times \frac{n(n+1)}{2}$, and if entries are the same, it is $m \times n$.

Principal Submatrix

A square submatrix of a square matrix $A_{n \times n}$ in which the principal diagonal entries are also principal diagonal entries of the original matrix.

Trace

The sum of the principal diagonal elements of a square matrix, denoted as $tr(A) = \sum_{i=1}^n a_{ii}$.

Similar Matrix

A matrix $M$ is similar to $N$ if there exists a non-singular matrix $P$ such that $P M P^{-1} = N$. Similar matrices share the same determinant, trace, and eigenvalues.

Adjoint of a Matrix

The transpose of the cofactor matrix of a square matrix $A$, denoted as $adj(A)$. Key property: $A \times adj(A) = |A| I$.

Symmetric Matrix

A square matrix $M$ such that its transpose is equal to itself ($M^T = M$).

Skew-Symmetric Matrix

A square matrix $M$ such that its transpose is equal to its negative ($M^T = -M$). The principal diagonal elements of a skew-symmetric matrix must be zero.

Orthogonal Matrix

A square matrix $M$ that satisfies $M M^T = I = M^T M$, meaning its inverse is equal to its transpose ($M^{-1} = M^T$) and its determinant is $\pm 1$.

Conjugate of a Matrix

A matrix $\bar{A}$ obtained by replacing every complex entry $a_{ij} = x + iy$ in matrix $A$ with its complex conjugate $\bar{a}_{ij} = x - iy$.

Tranjugate

The transpose of the conjugate of a matrix (or conjugate of the transpose), denoted as $A^{\theta}$. It is also known as the conjugate transpose.

Hermitian Matrix

A square matrix $M$ such that $M^{\theta} = M$. The elements on the principal diagonal must be real.

Skew-Hermitian Matrix

A square matrix $M$ such that $M^{\theta} = -M$. The principal diagonal elements are either zero or purely imaginary.

Unitary Matrix

A square matrix $M$ such that $M M^{\theta} = I = M^{\theta} M$. The determinant of a unitary matrix has a modulus of $1$.

Idempotent Matrix

A square matrix $M$ such that $M^2 = M$.

Involutory Matrix

A square matrix $M$ such that $M^2 = I$, meaning the matrix is its own inverse.

Nilpotent Matrix

A square matrix $A$ for which there exists a positive integer $m$ such that $A^m = 0$.

Index of Nilpotent Matrix

The smallest positive integer $k$ such that $A^k = 0$ for a nilpotent matrix $A$.

Rank of Matrix

The non-negative integer $r$ representing the order of the highest order non-zero minor in a matrix $M$. It is also the number of non-zero rows in its Echelon form.

Elementary Matrix

A matrix obtained from an identity matrix by performing a single elementary row or column transformation.

Characteristic Equation

The equation $|A - \lambda I| = 0$ used to find the eigenvalues (characteristic roots) of a square matrix $A$.

Algebraic Multiplicity (AM)

The number of times a specific eigenvalue $\lambda_i$ is repeated as a root of the characteristic equation.

Geometric Multiplicity (GM)

The number of linearly independent eigenvectors corresponding to a specific eigenvalue $\lambda$, calculated as $n - Rank(A - \lambda I)$.

Cayley-Hamilton Theorem

The theorem stating that every square matrix satisfies its own characteristic equation.

Minimal Polynomial

The smallest degree monic polynomial $M_A(x)$ such that $M_A(A) = 0$. It divides every polynomial that annihilates the matrix $A$.

Algebraic Structure

A non-empty set $G$ equipped with one or more binary operations, denoted as $(G, \ast_1, \ast_2)$, such as $(N, +)$.

Field

A non-empty set $F$ with two binary operations (addition and multiplication) that is an abelian group under addition, while the set excluding the additive identity is an abelian group under multiplication, satisfying distributive laws.

Vector Space

A non-empty set $V$ of vectors over a field $F$ of scalars that is an abelian group under vector addition and is closed under scalar multiplication, following specific distributive and associative axioms.