Matrix operations

5. Matrix operations#

5.1. Matrices as linear maps#

Very often, an \(m\times n\)-matrix \(A\) with real entries represents a linear map from the space \(\RR^n\) of vectors of length \(n\) to the space \(\RR^m\) of vectors of length \(m\). Similarly, a matrix with complex entries represents a linear map from \(\CC^n\) to \(\CC^m\). This map is defined by the matrix-vector product.

Definition 5.1

Given a real (resp. complex) \(m\times n\)-matrix \(A\), the linear map associated with \(A\) is the map that sends each vector \(\vv\in\RR^n\) to the vector \(A\vv\in\RR^m\) (resp. each vector \(\vv\in\CC^n\) to the vector \(A\vv\in\CC^m\)).

Note

Matrix multiplication is defined so that the product \(AB\) represents the linear map gotten by first applying \(B\) and then \(A\). In other words, we have

(5.1)#\[ (AB)\vv = A(B\vv) \]

Note that on the left we have one matrix multiplication and one matrix-vector multiplication, while on the right we just have two matrix-vector multiplications.

In quantum physics, matrices are very often square and have some additional properties that make them suitable as operators on a physical system or observables of such a system. We will come back to this in the section on Hilbert spaces and operators.

Many of the definitions below are specifically for square matrices. A first example is that of commutators, which play an extremely important role in quantum physics.

Definition 5.2 (Commutator)

Given two \(n\times n\)-matrices \(A\) and \(B\), the commutator of \(A\) and \(B\) is the \(n\times n\)-matrix

\[ [A,B]=AB-BA. \]

5.2. Transpose and adjoint#

Definition 5.3 (Transpose)

The transpose of an \(m\times n\)-matrix \(A\) is the \(n\times m\)-matrix \(A^\top\) defined by

\[ (A^\top)_{i,j}=A_{j,i}. \]

Definition 5.4 (Adjoint)

The adjoint of an \(m\times n\)-matrix \(A\) is the \(n\times m\)-matrix \(A^\dagger\) obtained by applying the complex conjugate and the transpose operation:

\[ (A^\dagger)_{i,j}=\overline{A_{j,i}}. \]

The adjoint of \(A\) is also known as the conjugate transpose or Hermitian transpose of \(A\). The adjoint is also sometimes used to refer to the adjugate of a matrix, or the tranpose of the cofactor matrix, but we will not use the term in this manner.

5.3. Trace, determinant and characteristic polynomial#

Definition 5.5 (Trace)

The trace of an \(n\times n\)-matrix \(A\) is the sum of the diagonal entries of \(A\):

\[ \tr A = \sum_{i=1}^n A_{i,i}. \]

The determinant of an \(n\times n\)-matrix \(A\) can be defined in various ways; you may have seen a different definition than the one below. We will make use of permutations.

We write \(S_n\) for the set of all permutations of \(\{1,\ldots,n\}\); then \(S_n\) has \(n!\) elements. To any \(\sigma\in S_n\) we can attach a sign \(\sign(\sigma)\in\{\pm1\}\). The sign of a permutation is \((-1)^{N(\sigma)}\), where \(N(\sigma)\) is the number of transpositions in the transposition decomposition of the permutation \(\sigma\).

Definition 5.6 (Determinant)

The determinant of an \(n\times n\)-matrix \(A\) is defined as

\[ \det A = \sum_{\sigma\in S_n} \sign(\sigma) \prod_{i=1}^n A_{i,\sigma(i)}. \]

Using the Einstein summation convention and the Levi-Cività symbol commonly used in physics, we can also write this as

\[ \det A=\epsilon^{i_1\cdots i_n} A_{1,i_1}\cdots A_{n,i_n}. \]

Note that except in small cases, this is usually not how you compute a determinant. One option is to expand along a row or column, but this is not much more efficient than directly using the definition. To compute the determinant of a larger matrices \(A\), it is most convenient to apply row (or column) operations to reduce \(A\) to triangular (or echelon) form and to use properties 4–6 below.

Property 5.1 (Properties of the determinant)

  1. A square matrix \(A\) is invertible if and only if \(\det(A)\) is non-zero.

  2. Given two \(n\times n\)-matrices \(A\) and \(B\), we have

    \[ \det(AB)=(\det A)(\det B). \]

  3. The determinant of a diagonal matrix (and more generally of an upper or lower triangular matrix) is the product of the diagonal entries.

  4. If \(B\) is obtained from \(A\) by scaling a row by a scalar \(c\), then \(\det B=c\det A\).

  5. If \(B\) is obtained from \(A\) by swapping two rows, then \(\det B=-\det A\).

  6. If \(B\) is obtained from \(A\) by adding a multiple of a row to another row, then \(\det B=\det A\).

Properties 4–6 also hold for columns instead of rows.

Finally, we define the characteristic polynomial of a square matrix; this will be used in Diagonalisation.

Definition 5.7 (Characteristic polynomial)

The characteristic polynomial of an \(n\times n\)-matrix \(A\) is the polynomial of degree \(n\) in a variable \(t\) defined by

\[ \chi_A(t) = \det(t\id-A). \]

Note

An alternative definition of the characteristic polynomial (which agrees with the one above up to a factor \((-1)^n\)) is

\[ \chi_A(t) = \det(A-t\id). \]

5.4. Exercises#

Exercise 5.1

Show that the definition of the matrix product \(AB\) is the only one for which (5.1) holds.

Exercise 5.2

Compute \([A,B]\) for \(A=\begin{pmatrix}1& 2\\0& 1\end{pmatrix}\) and \(B=\begin{pmatrix}1& 0\\-1& 1\end{pmatrix}\).

Exercise 5.3

Show that if \(A\) is an \(m\times n\)-matrix and \(B\) is an \(n\times m\)-matrix, then we have \(\tr(AB)=\tr(BA)\).

Exercise 5.4

Show that if \(A\) is an \(n\times n\)-matrix and \(c\) is a scalar, then

\[ \tr(cA)=c\tr A \quad\text{and} \det(cA)=c^n\det A. \]

Exercise 5.5

Compute the characteristic polynomial of the matrices

\[\begin{split} \begin{pmatrix}2& -1\\ 3& 2\end{pmatrix} \end{split}\]

and

\[\begin{split} \begin{pmatrix}1& 0& a\\ 0& -1& b\\ 1& 0& c\end{pmatrix} \end{split}\]

(where \(a\), \(b\), \(c\) are scalars).

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