Problems and Solutions in Introductory and Advanced Matrix Calculus

The following sections are included:

• Preface
• Note to the Readers
• Contents
• Notation

Let 𝔽 be a field, for example the set of real numbers ℝ or the set of complex numbers ℂ. Let m, n be two integers ≥ 1. An array A of numbers in 𝔽

Let A be an m× n matrix over a field 𝔽. Let b₁, … , b_m be elements of the field 𝔽. The system of equations

Let A be an m×n matrix and B be an r ×s matrix. The Kronecker product of A and B is defined as the (m · r) × (n · s) matrix

The Kronecker product is associative

A function on n × n matrices det : ℂ^n×n → ℂ is called a determinant function if and only if it satisfies the following conditions:

• det is linear in each row if the other rows of the matrix are held fixed.
• If the n × n matrix A has two identical rows then det(A) = 0.
• If I_n is the n × n identity matrix, then det(I_n) = 1.

Let A be an n×n matrix over ℂ. The complex number λ is called an eigenvalue of A if and only if the matrix (A − λI_n) is singular. Let x be a nonzero column vector in ℂⁿ. Then x is called an eigenvector belonging to (or associated with) the eigenvalue λ if and only if (A − λI_n)x = 0. The equation

An n × n matrix over ℂ is called normal if A*A = AA*. Let A be an n × n normal matrix over ℂ with eigenvalues λ₁, … , λ_n and corresponding pairwise orthonormal eigenvectors v_j (j = 1, … n). Then the matrix A can be written as (spectral decomposition)

Note that

Let A and B be n × n matrices. Then we define the commutator of A and B as

For all n × n matrices A, B, C we have the Jacobi identity

where 0_n is the n × n zero matrix. Since tr(AB) = tr(BA) we have

A matrix decomposition (or matrix factorization) is the right-hand side matrix product

Let p be a polynomial

Consider an n × n matrix A over ℂ and the polynomial

The Cayley-Hamilton theorem states that substituting the matrix A in the characteristic polynomial results in the n × n zero matrix, i.e.

Suppose A = (a_ij) and B = (b_ij) are two n × m matrices with entries in some fields. Then the Hadamard product (sometimes called the Schur product) is the entrywise product of A and B, that is, the m×n matrix A•B whose (i, j) entry is a_ijb_ij, i.e.

A linear space V is called a normed space, if for every v ∈ V there is associated a real number ||v||, the norm of the vector v, such that

Let A be an m × n matrix over ℂ. One defines the vec operator (vectorization) of A as

A square matrix M over ℂ is called normal if

A square matrix M over ℂ is called nonnormal if

Examples of normal matrices are: hermitian, skew-hermitian, unitary, projection matrices. Examples for nonnormal matrices are

An m × n matrix A is a binary matrix if a_jk ∈ {0, 1} for j = 1, … ,m and k = 1, … , n. A binary matrix is a matrix over the two-element field GF(2) = {0, 1}. In GF(2) = {0, 1} we have that −1 = 1, i.e. 1 + 1 = 0. Thus A + A = 0 for all binary matrices A, where 0 is the zero matrix. The special linear group and general linear group coincide over GF(2), GL_n(GF(2)) = SL_n(GF(2)). The set of invertible 2 × 2 binary matrices SL₂(GF(2)) is

Consider the 2 × 2 matrices

An n × n matrix U over ℂ is called a unitary matrix if

Thus UU* = I_n for a unitary matrix. If U and V are unitary n × n matrices, then UV is an n × n unitary matrix. The columns in a unitary matrix are pairwise orthonormal. Thus the column vectors in a unitary matrix form an orthonormal basis in ℂ_n. If v is a normalized (column) vector in ℂ_n, then Uv is also a normalized vector. The n×n unitary matrices form a group under matrix multiplication. Let K be a skew-hermitian matrix, then exp(K) is a unitary matrix. For any unitary matrix U we can find a skew-hermitian matrix K such that U = exp(K). Let H be an n×n hermitian matrix and U := exp(iK) is a unitary matrix. The eigenvalues of a unitary matrix are of the form e^iϕ with ϕ ∈ ℝ. Let A be an arbitrary n × n matrix over ℂ. Then under the map A ↦ UAU* the eigenvalues are preserved and thus the trace and the determinant of A…

A group G is a set of objects { a, b, c, … } (not necessarily countable) together with a binary operation which associates with any ordered pair of elements a, b in G a third element ab in G (closure). The binary operation (called group multiplication) is subject to the following requirements:

• There exists an element e in G called the identity element such that eg = ge = g for all g ∈ G.
• For every g ∈ G there exists an inverse element g⁻¹ in G such that gg⁻¹ = g⁻¹g = e.
• Associative law. The identity (ab)c = a(bc) is satisfied for all a, b, c ∈ G

A real Lie algebra L is a real vector space together with a bilinear map

Let n ≥ 3. The braid group ℬ_n of n strings (or group of n braids) has n − 1 generators {σ₁, σ₂, … , σ_n−1} (pairwise distinct) satisfying the relations

A graph consists of a non-empty set of points, called vertices (singular: vertex) or nodes, and a set of lines or curves, called edges, such that every edge is attached at each end to a vertex. We assume that V = { 1, 2, … , n }. A digraph (directed graph) is a diagram consisting of points, called vertices, joined by directed lines, called arcs. Thus each arc joins two vertices in a specified direction. To distinguish them from undirected graphs the edges of a digraph are called arcs…

A complex inner product vector space (also called pre-Hilbert space) is a complex vector space together with an inner product. An inner product vector space which is complete with respect to the norm induced by the inner product is called a Hilbert space. Two finite dimensional Hilbert spaces are considered. The Hilbert space ℂ^d with the scalar product

Let A be an n × n matrix over ℝ. Homogeneous systems of linear differential equations with constant coefficients are given by…

Consider the m × n matrix

Let A(t) be an n × n matrix

The following sections are included:

• Bibliography
• Index