Interactive Notes

Linear Algebra
for Quantum Computing

Vectors, complex numbers, inner products, Dirac notation, matrices and eigenvalues — everything needed to understand qubits from scratch.

Contents

Section 01

Vectors — what they really are

A vector is an ordered list of numbers. Think of it as an arrow in space: it has a direction and a length. In mathematics it is written as a column:

v = v₁ v₂ v₃ column vector with 3 components

A vector in 2D has 2 components, in 3D it has 3, and in general in N dimensions it has N. We work in 2D for now because it is easy to visualize — but everything generalizes.

When you write v = (3, 2), you are saying: "start at the origin, go 3 units right and 2 up". The vector is the arrow from (0,0) to (3,2).

Click on the plane to place a vector and read its components

Column notation (transpose)

We often write vectors compactly. The symbol ᵀ (transpose) means a row vector is read as a column:

v = (3, 2)ᵀ = 32 same thing, two ways of writing it

This will come in handy with quantum notation.

Ex 1.1Reading a vector

The vector $v = (-2, 3)ᵀ$ has vertical component equal to:

v₂ =

The first number is the x component (horizontal), the second is the y component (vertical).

Ex 1.2Vector addition

Vector addition is performed component by component:

12 + 3 -1 = 1+3 2+(-1) = 41

Calculate $(2, 5)ᵀ + (-1, 3)ᵀ$ :

Result = ( , )ᵀ

Section 02

Vector Space, Norm and Hilbert Space

What is a vector space?

A vector space is simply a set of vectors on which two operations are defined: addition and scalar multiplication. The set ℝ² (all pairs of real numbers) is a vector space, and so is ℂᴺ (vectors with N complex components).

Definition — Norm (length of a vector)

The norm of a vector v = (v₁, v₂, ..., vₙ) is its "geometric length":

‖v‖ = \sqrt( v₁² + v₂² + ... + vₙ² )

In 2D this is the Pythagorean theorem! If v = (3, 4)ᵀ, then ‖v‖ = √(9+16) = √25 = 5.

Click to place a vector — see the norm computed in real time

A vector with norm = 1 is called a unit vector or normalized vector. In quantum computing, all states are unit vectors (‖ψ‖ = 1). This ensures the probabilities sum to 1.

What is Hilbert Space?

A Hilbert Space is simply a vector space equipped with an additional structure called the inner product (covered in section 3). In practice, for quantum computing:

Hilbert space = space of complex vectors ℂᴺ + inner product.
For a qubit N=2, therefore the Hilbert space is ℂ² — the vectors have 2 complex components.

Don't be intimidated by the name. "Hilbert" is simply the mathematician who studied these structures. Physically, every quantum state is a unit vector in this space.

Ex 2.1Computing the norm

Calculate the norm of the following vectors:

a) $v = (3, 4)ᵀ$ → ‖v‖ = √(3² + 4²) = ?

b) $u = (1, 0)ᵀ$ → ‖u‖ = ?

‖v‖ = ‖u‖ =

Ex 2.2normalizing a vector

To make a unit vector, divide by its norm: $û = v / ‖v‖$

normalize $v = (3, 4)ᵀ$ . Write the first component as a fraction (e.g. 3/5):

û₁ = û₂ =

‖v‖ = 5 (computed above). Divide each component by 5: û₁ = 3/5, û₂ = 4/5. Check: (3/5)² + (4/5)² = 9/25 + 16/25 = 1 ✓

Section 03

Complex Numbers — conjugate and phase

A complex number z has two parts: a real part and an imaginary part:

z = a + jb a = real part Re(z) b = imaginary part Im(z) j = imaginary unit, with the property j² = -1

In physics and engineering j is used for the imaginary unit (mathematicians use i). You will see both in your notes — they mean the same thing.

visualization — the Argand Plane

Every complex number is a point in the plane (or an arrow from the origin). The horizontal axis is the real part, the vertical axis is the imaginary part. Click to explore.

Click to place a complex number — read modulus and phase in real time

The Complex Conjugate z*

The conjugate of z = a + jb is obtained simply by flipping the sign of the imaginary part:

z = a + jb z* = a - jb \leftarrow only the sign of j changes! (3+4j)* = 3 - 4j (2-5j)* = 2 + 5j (7)* = 7 \leftarrow pure reals: unchanged (jb)* = -jb \leftarrow pure imaginary: sign flips

Geometrically on the Argand plane, the conjugate is the reflection across the real axis.

z (blue) and z* (red) — reflection across the horizontal axis. Click to move z.

Modulus |z| — the "length" of a complex number

The modulus of z is its distance from the origin — always a real number ≥ 0:

|z| = \sqrt( a² + b² ) \leftarrow Pythagorean theorem! Example: z = 3+4j \to |z| = \sqrt(9+16) = \sqrt25 = 5

Squared modulus |z|² — fundamental in quantum

In quantum computing |z|² appears constantly. There is an elegant trick using the conjugate:

|z|² = z \cdot z* = (a+jb)(a-jb) = a² + b² Expanding: (a+jb)(a-jb) = a² - ajb + ajb - j²b² = a² - j²b² = a² - (-1)\cdotb² \leftarrow j² = -1 ! = a² + b² ✓

This is why |α|² always appears in quantum measurements: it is the measurement probability — a positive real number, perfect for a probability!

The phase φ

Every complex number has an angle (phase) relative to the real axis. It can be written in polar form:

z = |z| \cdot e jφ = |z| \cdot (cos φ + j\cdotsin φ) \leftarrow Euler's formula φ = arctan( b / a )

In quantum, the phase φ of the amplitudes does not change probabilities (|e^(jφ)|² = 1 always), but it is crucial for interference — the engine of quantum algorithms.

Ex 3.1Complex conjugate

Write the conjugate of these numbers (remember: only change the sign of j):

a) z = 5 + 2j b) z = −3 − j c) z = 4 (pure real)

a) z* = b) z* = c) z* =

Single rule: change the sign in front of j. If there is no j (real number), the conjugate is identical to the original number.

Ex 3.2Squared modulus |z|²

Calculate |z|² = a² + b² for:

a) z = 3 + 4j → 3² + 4² = ?

b) z = 1/√2 + j/√2 → (1/√2)² + (1/√2)² = ? (appears very often in quantum!)

a) |z|² = b) |z|² =

Ex 3.3Multiplying z · z* — the trick

Given z = 2 + 3j, verify that z · z* = |z|² by multiplying explicitly:

(2+3j)(2-3j) = 4 - 6j + 6j - 9j² = 4 - 9\cdot(?)

Substitute j² = −1, then add:

j² = z·z* =

j² = −1 for definition. Therefore −9j² = −9·(−1) = +9. Poi 4 + 9 = 13. Let's verify: |z|² = 2² + 3² = 4 + 9 = 13 ✓

Section 04

Inner Product — measuring "alignment"

The inner product (or dot product) of two vectors measures how "aligned" they are. For real vectors in ℝᴺ:

⟨u, v⟩ = u₁v₁ + u₂v₂ + ... + uₙvₙ

Geometrically: $u \cdot v = ‖u‖ \cdot ‖v‖ \cdot cos(θ)$ where θ is the angle between the two vectors.

Drag the slider to change the angle between the vectors — observe how the inner product changes

θ = 45°

Orthogonality

Two vectors are orthogonal (perpendicular) if their inner product is zero — the angle between them is 90°.

u \cdot v = 0 ⟺ u ⊥ v (orthogonal)

Inner product with complex numbers

When the components are complex numbers (as in quantum), the inner product is modified: the first vector must be conjugated:

⟨u, v⟩ = u₁*\cdotv₁ + u₂*\cdotv₂ + ... \leftarrow the * denotes complex conjugate

The conjugate of a complex number z = a + jb is z* = a − jb. Only the sign of the imaginary part changes. On the Argand plane this is the reflection across the real axis.

Ex 3.1Real inner product

Calculate $u \cdot v$ with $u = (2, 3)ᵀ$ and $v = (4, -1)ᵀ$

u · v = 2·4 + 3·(−1) = ?

u · v =

Ex 3.2Checking orthogonality

Are $a = (1, 1)ᵀ$ and $b = (1, -1)ᵀ$ orthogonal? Calculate the inner product:

a · b = Orthogonal?

Section 05

Dirac Notation — the language of Quantum

Quantum computing uses a special notation invented by Paul Dirac, which looks strange at first but is very convenient. Let's learn it step by step.

Ket |v⟩ — the column vector

The symbol |v⟩ (read "ket v") is simply a column vector. Nothing mysterious:

|ψ⟩ = α β \leftarrow just a regular column vector! Same as writing ψ = (α, β)ᵀ

Bra ⟨v| — the conjugate transpose

The symbol ⟨v| (read "bra v") is the conjugate transpose of the ket. That is: transpose the column to a row, and conjugate every complex component:

If |ψ⟩ = α β then ⟨ψ| = ( α* β* ) For real vectors: ⟨ψ| = ( α β ) \leftarrow transposed row

Bra + Ket = "Bracket". The product ⟨u|v⟩ = inner product of u and v. That is the whole idea!

Inner product in Dirac notation

⟨u|v⟩ = ( u₁* u₂* ) \cdot v₁ v₂ = u₁*\cdotv₁ + u₂*\cdotv₂ Identical to the inner product from the previous section.

Ex 4.1Writing bra and ket

Given $|ψ⟩ = (2, -3)ᵀ$ (real components),

a) The corresponding bra ⟨ψ| is a ___ vector (row/column)?

b) What is the second component of ⟨ψ|?

type: ⟨ψ|₂ =

Ex 4.2Computing ⟨u|v⟩

Given $|u⟩ = (1, 2)ᵀ$ and $|v⟩ = (3, 4)ᵀ$ (all real),

calculate ⟨u|v⟩ = (1, 2)·(3, 4)ᵀ = 1·3 + 2·4 = ?

⟨u|v⟩ =

Section 06

Why |0⟩ = (1, 0)ᵀ and |1⟩ = (0, 1)ᵀ?

This is a fundamental question. The short answer: it is a conventional choice, like choosing a coordinate system in geometry. Here is the full explanation.

The standard basis of ℝ²

In 2D, the "standard basis" vectors are the simplest ones pointing along the axes:

e₁ = 10 e₂ = 01 32 = 3\cdot 10 + 2\cdot 01 = 3\cdote₁ + 2\cdote₂

The connection to qubits

A qubit can be in one of two distinguishable and opposite states: "zero" and "one" (think: spin-up / spin-down, horizontal / vertical, etc.).

We choose to represent them with the standard basis vectors:

|0⟩ = 10 \leftarrow "spin up" |1⟩ = 01 \leftarrow "spin down"

The reason for this choice: |0⟩ and |1⟩ are orthogonal (⟨0|1⟩ = 0) and normalized (⟨0|0⟩ = 1). They form an orthonormal basis — the most convenient basis type in all of quantum computing.

Basis vectors |0⟩ and |1⟩ in the plane — orthogonal and unit length

Why these two vectors and not others?

We could choose any other pair of orthogonal unit vectors. For example:

|+⟩ = 1/\sqrt2 1/\sqrt2 |-⟩ = 1/\sqrt2 -1/\sqrt2 \to Hadamard basis — equally valid!

The choice of |0⟩=(1,0)ᵀ and |1⟩=(0,1)ᵀ is called the standard computational basis and is the most widely used by convention.

Ex 5.1Verifying the basis

Verify that |0⟩ and |1⟩ form an orthonormal basis:

a) ⟨0|0⟩ = (1, 0)·(1, 0)ᵀ = ?

b) ⟨1|1⟩ = (0, 1)·(0, 1)ᵀ = ?

c) ⟨0|1⟩ = (1, 0)·(0, 1)ᵀ = ? (orthogonal?)

⟨0|0⟩= ⟨1|1⟩= ⟨0|1⟩=

Ex 5.2Decomposition in the basis

Every vector can be written as a combination of |0⟩ and |1⟩. Write $v = (3, -2)ᵀ$ in ket notation:

v = ___ \cdot |0⟩ + ___ \cdot |1⟩

v = |0⟩ + |1⟩

α·|0⟩ + β·|1⟩ = α·(1,0)ᵀ + β·(0,1)ᵀ = (α, β)ᵀ. So α is the first component and β is the second.

Section 07

The Qubit — the basic quantum state

A qubit is the basic unit of quantum computing. Unlike a classical bit (which is only 0 or 1), a qubit can be in a superposition of |0⟩ and |1⟩.

Definition — Qubit

|ψ⟩ = α|0⟩ + β|1⟩ = α\cdot 10 + β\cdot 01 = α β α, β \in ℂ (complex numbers) |α|² + |β|² = 1 (unit norm)

α and β are called probability amplitudes.

Why |α|² + |β|² = 1?

This is exactly the normalization condition we studied. In Dirac terms:

⟨ψ|ψ⟩ = ( α* β* ) \cdot α β = |α|² + |β|² = 1

Physically: the total probability of finding the qubit in some state must be 1 (100%).

The unit circle — all real qubits lie on it. Click to place a state.

Qubit examples

|ψ⟩ = |0⟩ α=1, β=0 \to 1²+0² = 1 ✓ |ψ⟩ = |1⟩ α=0, β=1 \to 0²+1² = 1 ✓ (|0⟩+|1⟩)/\sqrt2 α=β=1/\sqrt2 \to 1/2+1/2 = 1 ✓ (|0⟩-|1⟩)/\sqrt2 α=1/\sqrt2, β=-1/\sqrt2 \to 1/2+1/2 = 1 ✓

The last example

(|0⟩+|1⟩)/\sqrt2

is the QRNG from your notes! Measuring this state gives 0 or 1 each with exactly 50% probability.

Ex 6.1Valid qubit or not?

For each one, check whether it is a valid qubit (norm = 1):

a) $|ψ⟩ = (1/2)|0⟩ + (\sqrt3/2)|1⟩$ → |1/2|² + |√3/2|² = ?

b) $|ψ⟩ = (1/2)|0⟩ + (1/2)|1⟩$ → valid?

a) sum = valid?

b) sum = valid?

a) (1/2)² + (√3/2)² = 1/4 + 3/4 = 4/4 = 1. b) (1/2)² + (1/2)² = 1/4 + 1/4 = 1/2 ≠ 1.

Ex 6.2Finding the missing β

Given $|ψ⟩ = (1/\sqrt3)|0⟩ + β|1⟩$ with β real and positive,

find β knowing it is a valid qubit.

Write the normalization condition: (1/√3)² + β² = 1

1/3 + β² = 1 → β² = 2/3 → β = ?

β =

Section 08

Measurement — where the quantum world meets the classical

If we have $|ψ⟩ = α|0⟩ + β|1⟩$ , what happens when we measure it?

Born Rule — Measurement Probability

P( outcome |φ⟩ ) = |⟨φ|ψ⟩|² P( |0⟩ ) = |⟨0|ψ⟩|² = |α|² P( |1⟩ ) = |⟨1|ψ⟩|² = |β|²

After measurement, the state collapses to the outcome obtained.

The symbol |·|² denotes the squared modulus. For real numbers |x|² = x². For complex z = a+jb, |z|² = a² + b². Always a real number ≥ 0 — perfect for a probability!

Measurement simulator — move the slider and press Measure

θ = 45° α=cos θ, β=sin θ

State collapse

After measuring and obtaining, say, |0⟩, the system's state becomes |0⟩. The next measurement gives 0 with certainty (probability 1). The system has "forgotten" the original amplitudes.

This is why quantum computing differs from classical computing: before measurement the system is in superposition (both 0 and 1 simultaneously), but measurement "forces" it to pick one.

Ex 7.1Computing probabilities

Dato $|ψ⟩ = (\sqrt3/2)|0⟩ + (1/2)|1⟩$

a) P(measure |0⟩) = |√3/2|² = ?

b) P(measure |1⟩) = |1/2|² = ?

c) Which outcome is more likely?

P(|0⟩)= P(|1⟩)= more likely:

Ex 7.2QRNG — from the notes

As in your notes: $|ψ⟩ = (|0⟩ + |1⟩)/\sqrt2$

a) Write α and β explicitly

b) Calculate P(|0⟩) and P(|1⟩)

c) Why is this a perfect random number generator?

α = β =

P(|0⟩)= P(|1⟩)=

α = 1/√2 ≈ 0.707. P(|0⟩) = (1/√2)² = 1/2 = 50%. Both the probabilities are equal → perfect randomness!

Section 09

Matrices and Quantum Gates — transforming qubits

How to read a matrix

A matrix is a grid of numbers organized in rows and columns. A 2×2 matrix has 2 rows and 2 columns:

Standard notation a b c d Compact notation ((a,b), (c,d)) or: [[a,b],[c,d]] Each inner pair = one row. First pair = row 1, second = row 2.

Example — gate X written compactly: (0 1 ; 1 0)
Read as: row 1 = (0, 1) · row 2 = (1, 0)
That is exactly:

X = ( 0  1 )  ← row 1: "0 1"

    ( 1  0 )  ← row 2: "1 0"

So (0 1 ; 1 0) and ((0,1),(1,0)) and the two-row form are three ways of writing the same thing.

Scaling to larger matrices — 3×3 and beyond

Same logic: each block separated by ";" is a row. A 3×3 matrix has 3 blocks:

Standard a b c d e f g h i Compact ((a,b,c), (d,e,f), (g,h,i)) Example — 3\times3 identity matrix: 100010001 ((1,0,0), (0,1,0), (0,0,1))

4×4 Matrix — two qubits

CNOT gate, the most common 2-qubit gate — 4 rows separated by ";":

CNOT gate — the most-used 2-qubit gate: 1000010000010010 ((1,0,0,0), (0,1,0,0), (0,0,0,1), (0,0,1,0))

Trick for reading compact notation: count ";" and add 1 → rows. Count numbers in the first block → columns. (1 2 3 ; 4 5 6) = 1 ";" → 2 rows, 3 numbers → 3 columns = 2×3 matrix.

Matrix-vector product

The rule: each row of the matrix takes the inner product with the column vector:

a b c d \cdot x y = a\cdotx + b\cdoty c\cdotx + d\cdoty \leftarrow row by components Example — X|0⟩: X|0⟩ = 0110 \cdot 10 = 0\cdot1+1\cdot0 1\cdot1+0\cdot0 = 01 = |1⟩

Unitary matrices — the quantum condition

In quantum computing, gates must be unitary matrices: UᴴU = I, where Uᴴ is the conjugate transpose of U.

Definition — Unitary Matrix

U is unitary if

Uᴴ \cdot U = I

where Uᴴ = (Uᵀ)* = transpose then conjugate each element.

Why? Because it preserves the vector norm: ‖U|ψ⟩‖ = ‖|ψ⟩‖ = 1. States remain valid states!

Gates from your notes — in both notations

Gate X (NOT) 0110 ((0,1),(1,0)) Gate Z (Phase flip) 10 0 -1 ((1,0),(0,-1)) Gate Y 0 -j j 0 ((0,-j),(j,0)) Gate H (Hadamard) 1/\sqrt2 \cdot 11 1 -1 1/\sqrt2\cdot((1,1),(1,-1))

Gate visualization — observe how it transforms the basis vectors

Ex 8.1Applying gate X

Calculate X|1⟩ by multiplying the matrix by the vector:

X|1⟩ = 0110 \cdot 01 = ?

X|1⟩ = ( , )ᵀ =

Ex 8.2Hadamard gate — superposition

Calculate H|0⟩ where $H = (1/\sqrt2)\cdot[[1,1],[1,-1]]$

First row × |0⟩ = (1/√2)·(1·1 + 1·0) = ?

Second row × |0⟩ = (1/√2)·(1·1 + (−1)·0) = ?

Result = (1/√2)|0⟩ + (1/√2)|1⟩ = ?

coeff. on |0⟩: coeff. on |1⟩:

Section 10

Eigenvalues and Eigenvectors — in matrix form

Definition — Eigenvector and Eigenvalue

A vector |u⟩ ≠ 0 is an eigenvector of matrix L with eigenvalue λ if:

L |u⟩ = λ |u⟩

In explicit matrix form:

det(L - λI) = 0 \leftarrow characteristic equation For a 2\times2 matrix: det a-λ b c d-λ = (a-λ)(d-λ) - b\cdotc = 0 Expanding: ad - aλ - dλ + λ² - bc = 0 λ² - (a+d)λ + (ad-bc) = 0 \leftarrow degree-2 polynomial

The matrix scales the vector by λ without changing its direction.

Generic vectors change direction; eigenvectors (coloured) stay on the same line — select the matrix

Step 1 — Rewrite as a homogeneous system

Start from L|u⟩ = λ|u⟩ and move everything to the left. Recall that λ|u⟩ = λI|u⟩ where I is the identity matrix:

Start from: L|u⟩ = λ|u⟩ Rearrange: L|u⟩ - λI|u⟩ = 0 \to (L - λI)|u⟩ = 0 In matrix form: l₁₁-λ l₁₂ l₂₁ l₂₂-λ \cdot u₁ u₂ = 00

Subtracting λI means subtracting λ only from the main diagonal, because I has 1 on the diagonal and 0 elsewhere:

( a  b )  −  λ( 1  0 )  =  ( a−λ   b  )

( c  d )      ( 0  1 )     (  c    d−λ )

Step 2 — Existence condition: det = 0

The system (L − λI)|u⟩ = 0 has a non-zero solution only if (L − λI) is singular (non-invertible). This occurs when the determinant is zero:

det(L - λI) = 0 \leftarrow characteristic equation For a 2\times2 matrix: det a-λ b c d-λ = (a-λ)(d-λ) - b\cdotc = 0 Expanding: ad - aλ - dλ + λ² - bc = 0 λ² - (a+d)λ + (ad-bc) = 0 \leftarrow polynomial of degree 2

For a matrix N×N the characteristic equation is a polynomial of degree N in λ → there are always N eigenvalues (counting multiplicities), possibly complex.

Full example — Pauli Z, all steps

Step 1 — Write the matrix:

Z = 10 0 -1

Step 2 — Calculate Z − λI by subtracting λ from the diagonal:

Z - λI = 10 0 -1 - λ \cdot 1001 = 1-λ 0 0 -1-λ

Step 3 — Calculate the determinant (diagonal matrix: product of diagonal elements):

det(Z - λI) = (1-λ)\cdot(-1-λ) Expanding: (1-λ)(-1-λ) = -1 - λ + λ + λ² = λ² - 1

Step 4 — Solve the characteristic equation:

λ² - 1 = 0 \to λ² = 1 \to λ = \pm1 \to λ₁ = +1 λ₂ = -1

Step 3 — Finding the eigenvectors

For each eigenvalue λₖ, substitute into (L − λₖI) and solve (L − λₖI)|u⟩ = 0:

Case λ₁ = +1 — substitute λ=+1:

1-1 0 0 -1-1 = 00 0 -2 \cdot u₁ u₂ = 00 row 1: 0\cdotu₁ + 0\cdotu₂ = 0 \to u₁ free row 2: -2\cdotu₂ = 0 \to u₂ = 0 Choose u₁=1 \to |u₁⟩ = 10 = |0⟩ ✓

Case λ₂ = −1 — substitute λ=−1:

1+1 0 0 -1+1 = 2000 \cdot u₁ u₂ = 00 row 1: 2\cdotu₁ = 0 \to u₁ = 0 row 2: 0 = 0 \to u₂ free Choose u₂=1 \to |u₂⟩ = 01 = |1⟩ ✓

Second example — Pauli X, non-trivial eigenvectors

Gate X has the same eigenvalues as Z (±1) but different and more interesting eigenvectors:

Steps 1–4 — Eigenvalues (computed above): λ₁ = +1, λ₂ = −1

Eigenvector for λ₁ = +1:

0-1 1 1 0-1 = -1 1 1 -1 \cdot u₁ u₂ = 00 row 1: -u₁ + u₂ = 0 \to u₂ = u₁ u₁=1, u₂=1. Normalize: |u₁⟩ = 1/\sqrt2 \cdot 11 = 1/\sqrt2 1/\sqrt2 = |+⟩

Eigenvector for λ₂ = −1:

0+1 11 0+1 = 1111 \cdot u₁ u₂ = 00 row 1: u₁ + u₂ = 0 \to u₂ = -u₁ Choose u₁=1, u₂=-1. Normalize: |u₂⟩ = 1/\sqrt2 \cdot 1 -1 = 1/\sqrt2 -1/\sqrt2 = |-⟩

The eigenvectors of X are |+⟩ and |−⟩ — the same states the Hadamard gate produces from |0⟩ and |1⟩! Measuring X means measuring in the "Hadamard basis" instead of the computational one.

Normalizing an eigenvector

An eigenvector found by solving the system has arbitrary components. To use it as a quantum state it must be normalized: divide by its norm.

|u⟩ = a b ‖|u⟩‖ = \sqrt(a²+b²) |û⟩ = 1/\sqrt(a²+b²) \cdot a b = a/\sqrt(a²+b²) b/\sqrt(a²+b²) Check: ⟨û|û⟩ = a²/(a²+b²) + b²/(a²+b²) = 1 ✓

The connection to quantum measurement

Every quantum observable is a Hermitian matrix L.
The possible measurement outcomes = eigenvalues of L (always real for Hermitian matrices).
After measurement the state collapses to the eigenvector corresponding to the outcome obtained.

Pauli Z → measures "is it |0⟩ or |1⟩?" → eigenvalues ±1, eigenvectors |0⟩, |1⟩
Pauli X → measures "is it |+⟩ or |−⟩?" → eigenvalues ±1, eigenvectors |+⟩, |−⟩

Ex 10.1Subtracting λI from the diagonal

Given the matrix $A = ((3, 1),(1, 3))$ , write A − λI in matrix form.

Remember: λ is subtracted only from the diagonal elements.

A − λI = (( , 1), (1, ))

Subtract λ only from the diagonal: element (1,1) = 3−λ, element (2,2) = 3−λ. Off-diagonal elements (1 and 1) remain unchanged.

Ex 10.2Determinant and eigenvalues of A = ((3,1),(1,3))

With A − λI = ((3−λ, 1),(1, 3−λ)), calculate the determinant:

det(A - λI) = (3-λ)(3-λ) - 1\cdot1 = (3-λ)² - 1 = 9 - 6λ + λ² - 1 = λ² - 6λ + 8 = 0

Solve with the quadratic formula: λ = (6 ± √(36−32)) / 2 = (6 ± ?) / 2

√(36−32) = λ₁ = λ₂ =

√(36−32) = √4 = 2. Then λ = (6±2)/2 → λ₁ = 8/2 = 4, λ₂ = 4/2 = 2. Or factorize: λ²−6λ+8 = (λ−4)(λ−2) = 0.

Ex 10.3Finding the eigenvector for λ₁ = 4

Substitute λ=4 in (A − λI)|u⟩ = 0:

3-4 1 1 3-4 = -1 1 1 -1 \cdot u₁ u₂ = 00

From row 1: −u₁ + u₂ = 0 → u₂ = u₁. Choose u₁=1, then normalize.

The normalized eigenvector |u₁⟩ = ?

|u₁⟩ = ( , )ᵀ

u₁=1, u₂=1. Norm = √(1²+1²) = √2. Divide: (1/√2, 1/√2)ᵀ ≈ (0.707, 0.707)ᵀ. Note: this is the same |+⟩ as gate X!

Ex 10.4Direct verification — L|u⟩ = λ|u⟩

Verify that |u₁⟩ = (1/√2, 1/√2)ᵀ is truly an eigenvector of A with λ=4, by computing A|u₁⟩:

A|u₁⟩ = 3113 \cdot 1/\sqrt2 1/\sqrt2 = 3/\sqrt2 + 1/\sqrt2 1/\sqrt2 + 3/\sqrt2 = 4/\sqrt2 4/\sqrt2

What is 4/√2 simplified? And is this equal to 4·(1/√2) = λ·|u₁⟩? (yes/no)

4/√2 = A|u₁⟩ = λ|u₁⟩?

Practical tricks to speed up calculations

Trick 1 — aI + bL has the same eigenvectors as L
If M = aI + bL, then M|u⟩ = aI|u⟩ + bL|u⟩ = a|u⟩ + bλ|u⟩ = (a + bλ)|u⟩.
Eigenvectors are unchanged; eigenvalues become a + bλ.

Example: A = ((3,1),(1,3)) = 3I + X → same eigenvectors as X (i.e. |+⟩ and |−⟩), eigenvalues 3±1 = 4 and 2.

Trick 2 — Diagonal matrices: eigenvalues = diagonal elements
If D = ((d₁, 0), (0, d₂)) then the eigenvalues are simply d₁ and d₂, and the eigenvectors are e₁=(1,0)ᵀ and e₂=(0,1)ᵀ. No calculation needed.

Example: Pauli Z = ((1,0),(0,−1)) → eigenvalues +1 and −1 at a glance, eigenvectors |0⟩ and |1⟩.

Trick 3 — Trace and determinant
For a 2×2 matrix the characteristic equation is always λ² − tr(L)·λ + det(L) = 0, where:
tr(L) = sum of diagonal elements det(L) = ad − bc
So λ₁ + λ₂ = tr(L) and λ₁ · λ₂ = det(L). You can verify eigenvalues without redoing all the multiplications.

Example: Pauli X = ((0,1),(1,0)) → tr=0, det=−1 → λ₁+λ₂=0 and λ₁·λ₂=−1 → must be +1 and −1 ✓

Trick 4 — Unitary matrices: eigenvalues have modulus 1
If U is unitary (UᴴU = I), all its eigenvalues satisfy |λ| = 1. In quantum this means eigenvalues always lie on the unit circle in the complex plane. For Pauli matrices (which are also Hermitian) they are real → so they are exactly ±1.

If you find an eigenvalue with |λ| ≠ 1 for a unitary matrix, you have made a calculation error.

Trick 5 — Eigenvectors of Hermitian matrices are orthogonal
If L = Lᴴ (Hermitian) and λ₁ ≠ λ₂, then the corresponding eigenvectors satisfy ⟨u₁|u₂⟩ = 0. No need to verify this: it is guaranteed by construction.

Example: ⟨+|−⟩ = (1/√2)(1/√2) + (1/√2)(−1/√2) = 1/2 − 1/2 = 0 ✓ (X is Hermitian, λ₁≠λ₂ → orthogonal)

Trick 6 — Once you find one eigenvector, the other is orthogonal to it
For a 2×2 Hermitian matrix, once the first eigenvector (a, b)ᵀ is found, the second is automatically (−b*, a*)ᵀ (normalized). So you only need to solve one of the two equations and get the other for free.

Example: found |+⟩ = (1/√2, 1/√2)ᵀ for X → the other is (−1/√2, 1/√2)ᵀ... which is |−⟩ up to a global phase (irrelevant in quantum).

0	−j
j	0

Linear Algebrafor Quantum Computing

Vectors — what they really are

Column notation (transpose)

Vector Space, Norm and Hilbert Space

What is a vector space?

What is Hilbert Space?

Complex Numbers — conjugate and phase

visualization — the Argand Plane

The Complex Conjugate z*

Modulus |z| — the "length" of a complex number

Squared modulus |z|² — fundamental in quantum

The phase φ

Inner Product — measuring "alignment"

Orthogonality

Inner product with complex numbers

Dirac Notation — the language of Quantum

Ket |v⟩ — the column vector

Bra ⟨v| — the conjugate transpose

Inner product in Dirac notation

Why |0⟩ = (1, 0)ᵀ and |1⟩ = (0, 1)ᵀ?

The standard basis of ℝ²

The connection to qubits

Why these two vectors and not others?

The Qubit — the basic quantum state

Why |α|² + |β|² = 1?

Qubit examples

Measurement — where the quantum world meets the classical

State collapse

Matrices and Quantum Gates — transforming qubits

How to read a matrix

Scaling to larger matrices — 3×3 and beyond

4×4 Matrix — two qubits

Matrix-vector product

Unitary matrices — the quantum condition

Gates from your notes — in both notations

Eigenvalues and Eigenvectors — in matrix form

Step 1 — Rewrite as a homogeneous system

Step 2 — Existence condition: det = 0

Full example — Pauli Z, all steps

Step 3 — Finding the eigenvectors

Second example — Pauli X, non-trivial eigenvectors

Normalizing an eigenvector

The connection to quantum measurement

Practical tricks to speed up calculations

Linear Algebra
for Quantum Computing