Matrix Theory Reference Sheet

Welcome to this comprehensive reference sheet of basic matrix theory. This page covers fundamental matrix definitions, operations, algebraic properties, and the properties of the determinant.

Definitions

To lay down a solid foundation, here are the definitions of fundamental types of matrices that frequently appear in linear algebra.

Matrix Type	Mathematical Condition	Description / Notes
Square Matrix		A matrix with the same number of rows and columns.
Diagonal Matrix	for	A square matrix where all off-diagonal entries are zero.
Identity Matrix		A diagonal matrix with ones on the main diagonal. Acts as the multiplicative identity.
Upper Triangular	for	A square matrix where all entries below the main diagonal are zero.
Lower Triangular	for	A square matrix where all entries above the main diagonal are zero.
Symmetric Matrix	()	A square matrix that is equal to its transpose.
Skew-Symmetric	()	A square matrix equal to its negative transpose; diagonal entries must be zero ().
Hermitian Matrix	()	A complex square matrix equal to its conjugate transpose (the complex analogue of a symmetric matrix).
Orthogonal Matrix	()	A real square matrix whose rows and columns are orthonormal vectors.
Unitary Matrix	()	A complex square matrix whose conjugate transpose is its inverse (the complex analogue of an orthogonal matrix).

Operations

Matrix operations are the foundational building blocks of linear algebra. The table below outlines the most common operations performed on matrices, along with their mathematical notations, definitions, and constraints.

Operation	Notation / Formula	Description	Constraints & Dimensions
Matrix Addition		Component-wise sum of two matrices.	Same dimensions:
Scalar Multiplication		Multiplies every element of the matrix by a scalar .
Matrix Multiplication		Standard dot-product of rows of and columns of .	Inner dimensions must match:
Transpose		Swaps rows and columns.
Conjugate Transpose	(or )	Transposes the matrix and takes the complex conjugate of each entry.
Matrix Inverse		The unique matrix that yields the identity matrix when multiplied with .	Square matrix () and non-singular ()
Hadamard Product		Element-wise multiplication of two matrices.	Same dimensions:
Kronecker Product		Tensor product creating a block matrix where each element of scales .
Trace		Sum of the elements on the main diagonal.	Square matrix ()

Properties

Matrix algebra differs significantly from standard scalar algebra, most notably because matrix multiplication is non-commutative. The following table summarizes the key algebraic properties of matrix addition, multiplication, transposes, and inverses.

Property Category	Mathematical Law / Identity	Description & Notes
Commutative Law	Addition: Multiplication: (In general)	Matrix addition is always commutative. However, matrix multiplication is strictly non-commutative ( is only true for specific commuting matrices).
Associative Law	Addition: Multiplication: Scalar:	Parentheses can be grouped freely for addition, multiplication, and scalar scaling.
Distributive Law	Left Distributive: Right Distributive: Scalar Distributive:	Matrix multiplication distributes over matrix addition from both the left and right sides.
Identity & Zero	Identity: Zero Addition: Zero Product: and	represents the identity matrix, and is the zero matrix of matching dimensions.
Transpose Properties	Involution: Sum: Scalar: Product:	Note the reversed order in the product transpose: .
Inverse Properties	Involution: Scalar: () Product: Transpose: Hermitian:	Appliable only to non-singular, square matrices. Note the reversed order in the product inverse: .
Hermitian Properties	Involution: Sum: Scalar: Product:	Applies to conjugate transposes over the complex field . is the complex conjugate of .

Determinant

The determinant is a scalar value computed from a square matrix that encodes crucial properties of the linear transformation represented by the matrix, such as its volume scaling factor and invertibility.

Fundamental Calculations

For small square matrices, the determinant is calculated as:

Matrix:
Matrix (Sarrus' Rule / Cofactor Expansion):
Higher Dimensions ( Matrices):
For , the determinant can be calculated recursively using Laplace (Cofactor) Expansion along any row or column :

where is the submatrix (minor) obtained by deleting the -th row and -th column of .

Alternatively, the general Leibniz Formula defines the determinant for any dimension:

where is the set of all permutations of , and is the signature of the permutation .

Core Properties of Determinants

Let and be square matrices, and let be a scalar.

Property	Mathematical Formula	Description / Notes
Multiplicative		Generalizes to:
Transpose		The determinant of a matrix equals the determinant of its transpose.
Conjugate Transpose		The determinant of the conjugate transpose is the complex conjugate of the determinant.
Inverse		Applies if is invertible ().
Scalar Multiplication		Scaling an matrix scales each of its rows.
Determinant of Powers		For any integer (and if is invertible).
Singularity & Invertibility		A square matrix is invertible if and only if its determinant is non-zero.
Diagonal & Triangular		If is diagonal, upper triangular, or lower triangular.
Orthogonal Matrix		If is real orthogonal ().
Unitary Matrix		If is complex unitary ().

Effects of Elementary Row Operations

When performing Gaussian elimination, the determinant changes predictably with each type of elementary row operation:

Row Interchanges (Swapping two rows):
Swapping any two rows multiplies the determinant by .
Row Scaling (Multiplying a row by a scalar ):
Multiplying a single row by a scalar multiplies the determinant by .
Row Addition (Adding a multiple of one row to another):
Adding a scalar multiple of one row to another row does not change the determinant.