Linear Algebra

Since I have already studied the basics of the necessary mathematical tools, for now I want to prioritize physical understanding. Thus, I will deepen my understanding of the mathematics to the level necessary to understand, apply, and critically question the physical framework under study. Purely mathematical proofs and developments can be left for when they become necessary or when I decide to delve deeper into the foundations.

For example, it is not necessary, at this point, to delve deeper into the proof of the spectral theorem for Hermitian matrices, since its proof is predominantly mathematical and does not, by itself, add a new physical interpretation to the problem. On the other hand, it is worth deepening mathematical results that establish relationships directly relevant to the physical theory, such as proving that two operators that have the same complete basis of eigenvectors commute, since this relationship allows us to understand mathematically the compatibility between observables.

However, I will be sharing here a purely mathematical review, an almost verbatim reproduction of this material (Delft University of Technology (2026)). A more complete review can be found aqui.

Basic Matrix Arithmetic¶

An $m\times n$-matrix is an array of $m$ rows and $n$ columns containing (usually real or complex) numbers called the entries (or coefficients) of the matrix. The entries of a matrix $A$ are denoted by $A_{i,j}$, $a_{ij}$ or similar, where the first index refers to the row and the second to the column. That is, the entries are laid out like

You will often see notation like $A=(a_{i,j})$ or $A=(a_{i,j})_{{1\le i\le m\atop 1\le j\le n}}$ to indicate how the entries are denoted and/or how many rows and columns the matrix has.

In this section we focus on the basic operations that can be performed with matrices.

Addition and scalar multiplication

Like vectors, matrices can be added or subtracted entrywise. They can also be multiplied entrywise by a given scalar. For example,

Special matrices

A few (types of) matrices occur very often and have their own names:

- the $m\times n$ zero matrix $\begin{pmatrix}0 & 0 & \cdots & 0\\ 0 & 0 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & 0 \end{pmatrix}$

- the $n\times n$ identity matrix $\begin{pmatrix}1 & 0 & \cdots & 0\\ 0 & 1 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & 1 \end{pmatrix}$

- diagonal matrices $\begin{pmatrix}a_{1,1} & 0 & \cdots & 0\\ 0 & a_{2,2} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & a_{n,n} \end{pmatrix}$

- upper triangular matrices $\begin{pmatrix}a_{1,1} & a_{1,2} & \cdots & a_{1,n}\\ 0 & a_{2,2} & \cdots & a_{2,n}\\ \vdots & \ddots & \ddots & \vdots\\ 0 & \cdots & 0 & a_{n,n} \end{pmatrix}$

- lower triangular matrices $\begin{pmatrix}a_{1,1} & 0 & \cdots & 0\\ a_{2,1} & a_{2,2} & \ddots & \vdots\\ \vdots & \vdots & \ddots & 0\\ a_{n,1} & a_{n,2} & \cdots & a_{n,n} \end{pmatrix}$

Inner product of vectors

Definition Given two vectors $\boldsymbol{v}=(v_{1},\ldots,v_{n})$ and $\boldsymbol{w}=(w_{1},\ldots,w_{n})$, the inner product of $\boldsymbol{v}$ and $\boldsymbol{w}$ is the scalar $\boldsymbol{v}\cdot\boldsymbol{w}=v_{1}w_{1}+\cdots+v_{n}w_{n}.$

Properties of the inner product: For vectors $\boldsymbol{v}$, $\boldsymbol{w}$, $\boldsymbol{x}$ (all of the same length) and scalars $c$ the following hold:

• $\boldsymbol{v}\cdot(\boldsymbol{w}+\boldsymbol{x})=\boldsymbol{v}\cdot\boldsymbol{w}+\boldsymbol{v}\cdot\boldsymbol{x}$

• $(\boldsymbol{v}+\boldsymbol{w})\cdot\boldsymbol{x}=\boldsymbol{v}\cdot\boldsymbol{x}+\boldsymbol{w}\cdot\boldsymbol{x}$

• $\boldsymbol{v}\cdot(c\boldsymbol{w})=c(\boldsymbol{v}\cdot\boldsymbol{w})=(c\boldsymbol{v})\cdot\boldsymbol{w}$

• $\boldsymbol{v}\cdot\boldsymbol{w}=\boldsymbol{w}\cdot\boldsymbol{v}$

Matrix-vector multiplication

Matrices can be multiplied by vectors.

Definition: Given a matrix $A$ as in 1 and a vector

$$\boldsymbol{v}=\begin{pmatrix}v_{1}\\ \vdots\\ v_{n} \end{pmatrix}\in\mathbb{R}^{n}$$

(3)

the (matrix-vector) product of $A$ and $\boldsymbol{v}$ is the vector

$$A\boldsymbol{v}=\begin{pmatrix}a_{1,1}v_{1}+\cdots+a_{1,n}v_{n}\\ \vdots\\ a_{m,1}v_{1}+\cdots+a_{m,n}v_{n} \end{pmatrix}\in\mathbb{R}^{m}.$$

(4)

Equivalently, we can write

Properties of matrix-vector multiplication: For matrices $A$ and $B$, vectors $\boldsymbol{v}$ and $\boldsymbol{w}$ and scalars $c$ the following hold whenever the expressions are defined:

• $A(\boldsymbol{v}+\boldsymbol{w})=A\boldsymbol{v}+A\boldsymbol{w}$

• $A(c\boldsymbol{v})=c(A\boldsymbol{v})$

• $(A+B)\boldsymbol{v}=A\boldsymbol{v}+B\boldsymbol{v}$

• $(cA)\boldsymbol{v}=c(A\boldsymbol{v})$ :::

Note: Here are two other useful ways to think about matrix-vector products. First, in terms of inner products: the vector $A\boldsymbol{v}$ consists of the inner products of each of the rows of $A$ with the vector $\boldsymbol{v}$. Namely, if

then

Second, in terms of linear combinations: the vector $A\boldsymbol{v}$ is a linear combination of the columns of $A$ where the coefficients are the entries of $\boldsymbol{v}$. Namely, if

then

Matrix multiplication

Two matrices $A$ and $B$ can be multiplied if the number of columns of $A$ equals the number of rows of $B$.

Definition: If $A$ is an $m\times n$-matrix and $B$ is an $n\times p$-matrix, then the (matrix) product of $A$ and $B$ is the $m\times p$-matrix $AB$ defined as follows: $(AB)_{i,k}=\sum^{n}_{j=1}A_{i,j}B_{j,k}\quad(1\le i\le m,1\le k\le p).$

The product $AB$ can be viewed as $p$ matrix-vector multiplications: if $B=(\boldsymbol{b}_{1}\ \boldsymbol{b}_{2}\ \cdots\boldsymbol{b}_{p})$, then

Properties of matrix multiplication: For matrices $A$, $B$ and $C$ and scalars $c$ the following hold whenever the expressions are defined:

• $A(B+C)=AB+AC$

• $(A+B)C=AC+BC$

• $(AB)C=A(BC)$

• $(cA)B=c(AB)=A(cB)$ :::

Warning: Matrix multiplication is not commutative! That is, in general we have

Matrix Operations¶

Very often, an $m\times n$-matrix $A$ with real entries represents a linear map from the space $\mathbb{R}^{n}$ of vectors of length $n$ to the space $\mathbb{R}^{m}$ of vectors of length $m$. Similarly, a matrix with complex entries represents a linear map from $\mathbb{C}^{n}$ to $\mathbb{C}^{m}$. This map is defined by the matrix-vector product.

Definition: Given a real (resp. complex) $m\times n$-matrix $A$, the linear map associated with $A$ is the map that sends each vector $\boldsymbol{v}\in\mathbb{R}^{n}$ to the vector $A\boldsymbol{v}\in\mathbb{R}^{m}$ (resp. each vector $\boldsymbol{v}\in\mathbb{C}^{n}$ to the vector $A\boldsymbol{v}\in\mathbb{C}^{m}$). :::

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

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. Commutators play an extremely important role in quantum physics.

Definition: Given two $n\times n$-matrices $A$ and $B$, the commutator of $A$ and $B$ is the $n\times n$-matrix $[A,B]=AB-BA.$

Definition: 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: The adjoint (conjugate transpose or Hermitian transpose) 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 ** or ** of $A$.

Definition: The trace of an $n\times n$-matrix $A$ is the sum of the diagonal entries of $A$: $\text{Tr}\left(A\right)=\sum^{n}_{i=1}A_{i,i}.$

The determinant of an $n\times n$-matrix $A$ can be defined in various ways. We will make use of permutations. Using the Einstein summation convention^[1] and the Levi-Civita symbol^[2] commonly used in physics, we can also write this as

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

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

6. 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: The characteristic polynomial of an $n\times n$-matrix $A$ is the polynomial of degree $n$ in a variable $t$ defined by $\chi_{A}(t)=\det(x\mathbb{I}-A).$^[3]

Properties that are typically demonstrated in exercises:

• If $A$ is an $m\times n$ matrix and $B$ an $n\times m$ matrix, then $\text{Tr}\left(AB\right)=Tr\left(BA\right)$

• If $A$ is an $n\times n$ matrix and $c$ a scalar, then $\text{Tr}\left(cA\right)=c\text{Tr}\left(A\right)$ and $\det\left(ca\right)=c^{n}\det\left(A\right)$

Basis transformations¶

Until now, we have been working with one specific coordinate system in which vectors are identified with their coordinate vectors with respect to the standard basis vectors:

In other words, a vector $\boldsymbol{v}=\begin{pmatrix}v_{1}\\ v_{2}\\ \vdots\\ v_{n} \end{pmatrix}$ means the same as the corresponding linear combination of the standard basis vectors:

In many situations, it is useful to express vectors in a different coordinate system. For this we need the general concept of a basis.

Definition: A basis of $\mathbb{R}^{n}$ consists of $n$ vectors $(\boldsymbol{b}_{1},\ldots,\boldsymbol{b}_{n})$ (or $\left\{ \boldsymbol{b}_{i}\right\} ^{n}_{i=1}$) such that every vector $\boldsymbol{v}\in\mathbb{R}^{n}$ can be written in exactly one way as a linear combination:$\boldsymbol{v}=c_{1}\boldsymbol{b}_{1}+\cdots+c_{n}\boldsymbol{b}_{n}$, with scalars $c_{1},\ldots,c_{n}$. These scalars are called the coordinates of $\boldsymbol{v}$ relative to $\left\{ \boldsymbol{b}_{i}\right\} ^{n}_{i=1}$. The vector $\left(\begin{array}{ccc} c_{1} & \dots & c_{n}\end{array}\right)^{T}$ is called the coordinate vector of $\boldsymbol{v}$ relative to $\left\{ \boldsymbol{b}_{i}\right\} ^{n}_{i=1}$.

Alternatively, a basis consists of $n$ vectors $\left\{ \boldsymbol{b}_{i}\right\} ^{n}_{i=1}$ with the following two properties:

• they span the space $\mathbb{R}^{n}$, i.e. every vector is a linear combination of $\left\{ \boldsymbol{b}_{i}\right\} ^{n}_{i=1}$;

• they are linearly independent, i.e. the only way to write the zero vector as a linear combination of $\left\{ \boldsymbol{b}_{i}\right\} ^{n}_{i=1}$is as $0\boldsymbol{b}_{1}+\cdots+0\boldsymbol{b}_{n}$.

A priori, we could have defined a basis as a collection of some number (say $m$) of vectors that span the space and are linearly independent. However, it can be shown that in fact any two bases contain the same number of vectors. Since the standard basis of $\mathbb{R}^{n}$ consists of $n$ vectors, the same therefore holds for any basis.

Suppose we have two bases of $\mathbb{R}^{n}$, say

Definition: The basis transformation matrix (or change-of-basis matrix) from $B$ to $C$ is the matrix $P=P_{B\to C}$ such that the $j$-th column of $P$ contains the coordinates of $\boldsymbol{b}_{j}$ relative to the basis $(\boldsymbol{c}_{1},\ldots,\boldsymbol{c}_{n})$.

The definition of $P$ means:

Given a vector $\boldsymbol{x}$, we write $\boldsymbol{x}^{(B)}=(x^{(B)}_{1},\ldots,x^{(B)}_{n})$ for the coordinate vector of $\boldsymbol{x}$ with respect to the basis $\left\{ \boldsymbol{b}_{i}\right\} ^{n}_{i=1}$, and we define $\boldsymbol{x}^{(C)}$ similarly. Then we calculate

which shows that

Similarly, given a linear map $A$, we write $A^{(B)}$ for the matrix of $A$ with respect to the basis $\left\{ \boldsymbol{b}_{i}\right\} ^{n}_{i=1}$, and likewise for $A^{(C)}$. Then $A^{(B)}$ and $A^{(C)}$ are related by

Namely, consider what happens when we multiply this matrix to a coordinate vector of the form $\boldsymbol{x}^{\left(C\right)}$. First, applying $P^{-1}$^[4] to $\boldsymbol{x}^{\left(C\right)}$ gives $\boldsymbol{x}^{\left(B\right)}$. Next, applying $A^{(B)}$ gives $(A\boldsymbol{x})^{(B)}$. Finally, applying $P$ gives $(A\boldsymbol{x})^{(C)}$, which is what we wanted to show.

The trace and the determinant are invariant under basis transformation: if $A$ and $P$ are square matrices of the same size with $P$ invertible, then we have

Because the characteristic polynomial is also defined as a determinant, it too is invariant under basis transformation.

Properties that are typically demonstrated in exercises:

• Suppose that $A$ and $B$ are two $n\times n$ matrices and $P$ is an invertible $n\times n$ matrix such that $B=PAP^{-1}$, then $\text{Tr}\left(B\right)=\text{Tr}\left(A\right)$ and $\det B=\det A$, in addition, $A$ and $B$ share the same characteristic polynomial.

Diagonalisation¶

A very common example of a basis transformation is diagonalisation of a matrix. For this we first introduce the concept of eigenvalues and eigenvectors. Consider an $n\times n$-matrix $T$ with complex entries. We view $T$ as a linear map on $\mathbb{C}^{n}$.

Definition: An eigenvector for $A$ is a non-zero vector $\boldsymbol{v}$ such that $A\boldsymbol{v}=\lambda\boldsymbol{v}$ for some scalar $\lambda$. This $\lambda$ is called the eigenvalue of $T$ acting on $\boldsymbol{v}$.

Writing $I$ for the $n\times n$ identity matrix, we see that if $\boldsymbol{v}$ is an eigenvector with eigenvalue $\lambda$, then

This means that $A-\lambda I$ is not invertible^[5]; equivalently, its determinant vanishes. Thus, if $\lambda$ is an eigenvalue of $A$, we have

This implies that the eigenvalues of $A$ are roots of the characteristic polynomial $\chi_{A}$ . We can factor the characteristic polynomial as

where $\lambda_{1},\ldots,\lambda_{r}$ are the distinct roots of $\chi_{A}(t)$ and each $k_{i}$ is an integer called the multiplicity of $\lambda_{i}$ as a root of $\chi_{A}(t)$.

The idea of diagonalisation is that it is often easiest to understand a linear map $A$ if we can find a basis of our space consisting of vectors on which $A$ acts as multiplication by some scalar.

Definition: An $n\times n$-matrix $A$ is diagonalisable if there exist a diagonal^[6] matrix $D$ and an invertible $n\times n$-matrix $P$ such that $A=PDP^{-1},$

We can rewrite the above equation as $AP=PD$. Let us write $\boldsymbol{v}_{1},\ldots,\boldsymbol{v}_{n}$ for the columns of $P$ and $\lambda_{1},\ldots,\lambda_{n}$ for the diagonal entries of $D$. Then we compute

and

We therefore see that the equation $AP=PD$ comes down to the $n$ equations $A\boldsymbol{v}_{i}=\lambda_{i}\boldsymbol{v}_{i}$. In other words, each $\boldsymbol{v}_{i}$ is an eigenvector of $A$ and $\lambda_{i}$ is the corresponding eigenvalue.

The fact that $P$ is invertible means that the eigenvectors $\boldsymbol{v}_{i}$ form a basis of $\mathbb{C}^{n}$. We obtain the following conclusion.

Theorem: An $n\times n$-matrix $A$ is diagonalisable precisely when there exists a basis of $\mathbb{C}^{n}$ consisting of eigenvectors for $A$.

This also gives us a way to tell whether $A$ is diagonalisable, and if so to determine matrices $D$ and $P$ as above:

1. Determine the eigenvalues $\lambda_{1},\ldots,\lambda_{r}$ of $A$ by finding the roots of the characteristic polynomial $\chi_{A}$.

2. For each eigenvalue $\lambda_{i}$, say with multiplicity $k_{i}$, determine if there exist $k_{i}$ linearly independent eigenvectors $\boldsymbol{v}_{\lambda_{i},1},\ldots,\boldsymbol{v}_{\lambda_{i},k_{i}}$ with eigenvalue $\lambda_{i}$.

3. The matrix $A$ is diagonalisable if and only if there exist $k_{i}$ linearly independent eigenvectors with eigenvalue $\lambda_{i}$ for all $i$. In this case, the matrices

and

give a diagonalisation of $A$.

Properties that are typically demonstrated in exercises:

• Assuming $A=PDP^{-1}$ with $P$ invertible and $D$ a scalar matrix^[7], then $A=D$.

Hilbert Spaces¶

So far, we have been considering vectors in an $n$-dimensional real or complex space. In quantum mechanics, one encounters infinite-dimensional vector spaces as well. In particular, the state of a quantum system is represented mathematically by a (unit) vector in a Hilbert space.

Vector spaces¶

As is customary in quantum physics and quantum algorithms, we use bra-ket notation for vectors in Hilbert spaces.

Definition: A (real or complex) *vector space* consists of

- a set of vectors, denoted by symbols like $\left|\alpha\right\rangle$

- a distinguished vector $\mathbf{0}$ called the zero vector

- a way of adding vectors, i.e. given two vectors $\left|\alpha\right\rangle$ and $\left|\beta\right\rangle$ we can obtain a third vector $\left|\alpha\right\rangle +\left|\beta\right\rangle$

- a way of multiplying a vector by a scalar: given a scalar $c$ (a real or complex number, depending on whether we consider a real or complex vector space) and a vector $\left|\alpha\right\rangle$ we can obtain a vector $c\left|\alpha\right\rangle$

Such that a number of properties hold:

- $0\left|\alpha\right\rangle =\mathbf{0}$

- $1\left|\alpha\right\rangle =\left|\alpha\right\rangle$

- $\mathbf{0}+\left|\alpha\right\rangle =\left|\alpha\right\rangle$

- $\left|\alpha\right\rangle +\left|\beta\right\rangle =\left|\beta\right\rangle +\left|\alpha\right\rangle$

- $(\left|\alpha\right\rangle +\left|\beta\right\rangle )+\left|\gamma\right\rangle =\left|\alpha\right\rangle +(\left|\beta\right\rangle +\left|\gamma\right\rangle )$

- $c(\left|\alpha\right\rangle +\left|\beta\right\rangle )=c\left|\alpha\right\rangle +c\left|\beta\right\rangle$

- $(c+d)\left|\alpha\right\rangle =c\left|\alpha\right\rangle +d\left|\alpha\right\rangle$

- $c(d\left|\alpha\right\rangle )=(cd)\left|\alpha\right\rangle$

Definition: A basis for a vector space is a collection of vectors $\left\{ \left|e_{i}\right\rangle \right\} _{i}$ such that every vector can be written in exactly one way as a linear combination of the $\left\{ \left|e_{i}\right\rangle \right\} _{i}$. The number of vectors in a basis is called the dimension of the space.

As in basis-transformations, any two bases contain the same number of vectors, so the dimension does not depend on the choice of basis.

Exemplo: The simplest examples are the $n$-dimensional vector spaces $\mathbb{R}^{2}$ and $\mathbb{C}^{n}$ with coordinatewise addition and scalar multiplication. An important special case is the two-dimensional space $\mathbb{C}^{2}$. In bra-ket notation, various symbols are used for the standard basis vectors $\binom{1}{0}$ and $\binom{0}{1}$ in this space, such as:

• $\left|+\right\rangle$ and $\left|0\right\rangle$;

• $\left|0\right\rangle$ and $\left|1\right\rangle$ (in the context of quantum computing);

• $\left|\uparrow\right\rangle$ and $\left|\downarrow\right\rangle$ (in the context of spins).

Hermitian inner products and Hilbert spaces¶

Definition: A Hermitian inner product assigns to vectors $\left|\alpha\right\rangle$ and $\left|\beta\right\rangle$ a complex number $\left\langle \alpha|\beta\right\rangle$ such that the following properties hold:

- $\left\langle \alpha|\beta\right\rangle =\overline{\left\langle \beta|\alpha\right\rangle }$;

- $\left\langle \alpha|\alpha\right\rangle \ge0$, and $\left\langle \alpha|\alpha\right\rangle =0$ holds precisely when $\left|\alpha\right\rangle =\mathbf{0}$ (note that $\left\langle \alpha|\alpha\right\rangle$ is a real number because of the first property);

- $\left\langle \alpha\right|\left(\left|\beta\right\rangle +\left|\gamma\right\rangle \right)=\left\langle \alpha|\beta\right\rangle +\left\langle \alpha|\gamma\right\rangle$;

- $\left\langle \alpha\right|\left(c\left|\beta\right\rangle \right)=c\left(\left\langle \alpha|\beta\right\rangle \right)$;

Definition: The norm or length of a vector $\left|\alpha\right\rangle$ in a Hilbert space is defined as the real number $\|\alpha\|=\sqrt{\left\langle \alpha|\alpha\right\rangle }$.

The properties above imply that for any scalar $c$, the inner product of $c\left|\alpha\right\rangle$ with $\left|\beta\right\rangle$ equals $\bar{c}\left\langle \alpha|\beta\right\rangle$. In other words, while the inner product is (complex) linear in its second argument, it is only conjugate linear in its first argument. This is the usual convention in physics. It differs from the convention in mathematics, where inner products are usually linear in the first argument and conjugate linear in the second argument.

Definition: A Hilbert space is a complex vector space $\mathcal{H}$ equipped with a Hermitian inner product $\left\langle \enspace|\enspace\right\rangle$ and satisfying a condition called completeness.

We will not make the notion of completeness precise in these notes. Roughly speaking, it means that we do not have to worry too much about convergence of sequences or series in our Hilbert space, and there is a notion of infinite linear combinations^[8].

Example: We take $\mathcal{H}=\mathbb{C}^{n}$ and define an inner product by $\left\langle \alpha|\beta\right\rangle =\sum^{n}_{i=1}\bar{\alpha}_{i}\beta_{i}.$ This satisfies all the properties of a Hermitian inner product.

The dual Hilbert space¶

With any Hilbert space $\mathcal{H}$, we can associate a dual Hilbert space $\mathcal{H}^{*}$. In the bra-ket formalism used in quantum mechanics, for every “ket vector” $\left|\alpha\right\rangle$ in $\mathcal{H}$ we have a dual “bra vector” $\left\langle \alpha\right|$ in $\mathcal{H}^{*}$. This correspondence has the following properties:

In mathematical language, $\mathcal{H^{*}}$ is the space of continuous linear maps $\mathcal{H}\to\mathbb{C}$. The bra vector $\left\langle \alpha\right|$ then represents the linear map that sends $\left|\beta\right\rangle$ to $\left\langle \alpha|\beta\right\rangle$.

Example:In the case $\mathcal{H}=\mathbb{C}^{n}$ we can identify $\mathcal{H}$ with $\mathbb{C}^{n}$ as well; for a vector $\left|\alpha\right\rangle$ in $\mathbb{C}^{n}$, the dual vector $\left\langle \alpha\right|$ can then be naturally identified with the corresponding vector in $\mathbb{C}^{n}$. It can also be convenient to distinguish $\mathcal{H}$ from $\mathcal{H}^{*}$ by denoting a ket vector $\left|\alpha\right\rangle$ as a column vector and the corresponding bra vector $\left\langle \alpha\right|$ by the conjugate transpose of $\left|\alpha\right\rangle$, so $\left\langle \alpha\right|$ is the row vector whose entries are the complex conjugates of those of $\alpha$.

In $\mathbb{C}^{2}$, for example, we have

This allows us to view $\left|\alpha\right\rangle$ as a linear map sending $\left\langle \beta\right|$ to the scalar obtained by multiplying the row vector $\left|\alpha\right\rangle$ by the column vector $\left\langle \beta\right|$. If you know about continuous linear maps of Hilbert spaces, here is a useful connection between the mathematical definition of $\mathcal{H}^{*}$ and the bra-ket formalism: the Riesz representation theorem states that every continuous linear map $\mathcal{H}\to\mathbb{C}$ is of the form $\left\langle \alpha|\enskip\right\rangle$ for some $\alpha$ in $\mathcal{H}$. Ou seja, todo mapa linear contínuo $f:\mathcal{H}\to\mathbb{C}$ tem a forma $\left\langle \alpha|\enskip\right\rangle$ onde $\left\langle \alpha\right|\in\mathcal{H}^{*}$ tem um único vetor correspondente $\left|\alpha\right\rangle \in\mathcal{H}$.

Orthonormal bases¶

When working with vector spaces, it is frequently useful to make a choice of basis adapted to the problem at hand. In the context of a Hilbert space, it turns out that a “good” basis is often one that satisfies the following property with respect to the inner product.

Definition: An orthonormal system of vectors in a Hilbert space $\mathcal{H}$ is a collection of vectors $\left\{ \left|e_{i}\right\rangle \right\} _{i\in I}$, where $I$ is some index set, satisfying

$$\left\langle e_{i}|e_{j}\right\rangle =\begin{cases} 1 & \quad\text{if }i=j,\\ 0 & \quad\text{if }i\ne j. \end{cases}$$

(31)

Definition: An orthonormal basis of a Hilbert space $\mathcal{H}$ is an orthonormal system $\left\{ \left|e_{i}\right\rangle \right\} _{i\in I}$ in $\mathcal{H}$ that spans $\mathcal{H}$, i.e. every vector in $\mathcal{H}$ can be written as a (possibly infinite) linear combination of the $\left\{ \left|e_{i}\right\rangle \right\} _{i\in I}$.

Example: In the Hilbert space $\mathbb{C}^{n}$ with the standard inner product, the standard basis $(e_{1},\ldots,e_{n})$ is an orthonormal basis.

Linear subspaces¶

Definition: A linear subspace of a vector space $V$ is a set of vectors $W$ contained in $V$ such that the following properties hold:

- The zero vector of $V$ lies in $W$.

- For all vectors $w$ and $w'$ in $W$, the sum $w+w'$ is also in $W$.

- For every vector $w$ in $W$ and every scalar $\lambda$, the scalar multiple $\lambda w$ is also in $W$.

In quantum physics, linear subspaces are important because they encode quantum states that a physical state can collapse to when a measurement is performed. To make this more precise, we need the notions of orthogonal complement of a subspace and of orthogonal projection onto a subspace.

Definition: Orthogonal complement Consider a Hilbert space $\mathcal{H}$ and a linear subspace $W$ of $\mathcal{H}$. The orthogonal complement of $W$ in $\mathcal{H}$, denoted by $W^{\perp}$, is the set of all vectors $\left|v\right\rangle$ in $\mathcal{H}$ that are orthogonal to all of $W$, i.e.

It is not hard to verify that the orthogonal complement of a linear subspace of $\mathcal{H}$ is again a linear subspace.

Properties that are typically demonstrated in exercises:

• The Cauchy-Schwarz inequality states that for the vectors $\left|\alpha\right\rangle$ and $\left|\beta\right\rangle$, in a Hilbert space, we have $\left|\left\langle \alpha|\beta\right\rangle \right|^{2}\leq\left\langle \alpha|\alpha\right\rangle \left\langle \beta|\beta\right\rangle$

• The triangle inequality states that for the vectors $\left|\alpha\right\rangle$ and $\left|\beta\right\rangle$, in a Hilbert space, we have $\left\Vert \left|\alpha\right\rangle +\left|\beta\right\rangle \right\Vert \leq\left\Vert \left|\alpha\right\rangle \right\Vert +\left\Vert \left|\beta\right\rangle \right\Vert$

• If we consider $W$ a linear subspace of the Hilbert space $\mathcal{H},$ then the orthogonal complement^[9] $W^{\perp}$ is also a linear space.

Operators on Hilbert spaces¶

An operator or linear map on a Hilbert space $\mathcal{H}$ is a map

such that

• $T(c\left|\alpha\right\rangle )=c(T\left|\alpha\right\rangle )$

• $T(\left|\alpha\right\rangle +\left|\beta\right\rangle )=T\left|\alpha\right\rangle +T\left|\beta\right\rangle$

It is also possible to define linear maps between different Hilbert spaces, but we will not go into this.

In physics, operators are often denoted with a hat, so what we call $T$ would be written as $\hat{T}$. One reason for this is to distinguish a physical quantity (like momentum or energy) from the corresponding mathematical operator. Since we do not consider physical quantities in this module, we do not use the hat notation.

Operators form a vector space: the sum of two operators $T$ and $U$ is defined by the formula

and multiplication of an operator $T$ by a scalar $c$ is defined by the formula

Like matrices (but unlike vectors), operators can be composed: if $T$ and $U$ are two operators, then $TU$ is the operator defined by

Matrix entries¶

In physics, quantities of the form

are often of special interest. These are called matrix entries of $T$, especially if $\left|\alpha\right\rangle$ and $\left|\beta\right\rangle$ are basis vectors in some fixed orthonormal basis of $\mathcal{H}$.

A remark on notation: you will often see notation like $\left\langle \alpha|T\beta\right\rangle$. This is literally meaningless: the operator $T$ can be applied to vectors, which we denote by $\left|\beta\right\rangle$, while the symbol $\beta$ is just a label. However, in practice it is very convenient and completely unambiguous to write $\left\langle \alpha|T\beta\right\rangle$ instead of the strictly speaking more correct $\left\langle \alpha\right|(T\left|\beta\right\rangle )$.

Similarly, we will write $\left\langle T\alpha|\beta\right\rangle$ to mean the inner product of $T\left|\alpha\right\rangle$ with $\left|\beta\right\rangle$. Extending this notation to scalars $c$, we can also write

The adjoint of an operator¶

Given an operator $T$ on a Hilbert space $\mathcal{H}$, there exists an operator $T^{\dagger}$ with the property

The operator $T^{\dagger}$ is called the *adjoint* of $T$.

Properties of the adjoint: For all operators $T$, $U$ and scalars $c$, we have

1. $(cT)^{\dagger}=\bar{c}T^{\dagger}$

2. $(TU)^{\dagger}=U^{\dagger}T^{\dagger}$