Tensors

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Vectors

A vector is an ordered list of numbers. They can represent quantities with both magnitude and direction.

Example of a 2D vector: $\vec{v} = (3, 4)$

Operations

Addition: $(a,b) + (c,d) = (a+c, b+d)$
Subtraction: $(a,b) - (c,d) = (a-c, b-d)$
Scalar multiplication: $k(a,b) = (ka, kb)$
Dot product: $\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2$
Cross product (3D): $$\vec{u} \times \vec{v} = (u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1)$$

a + b

a − b

Drag the colored dots to move vectors.

Parallelogram Show −b helper Snap to grid

Matrices

A matrix is a rectangular array of numbers arranged in rows and columns.

Example: $$ A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix} $$

Operations

Addition: element-wise
Multiplication: row by column rule
Transpose: $A^T$
Determinant: $\det(A)$
Inverse: $A^{-1}$, when it exists

Eigenvalues and Eigenvectors

They satisfy: $$A \vec{v} = \lambda \vec{v}$$

Eigenvalues are central in many algorithms. For example, they underpin techniques like the Variational Quantum Eigensolver (VQE).

Tensors

Tensors generalize vectors (1D) and matrices (2D) to higher dimensions.

Tensors are fundamental to understanding data structures in computing and physics. In quantum computing, they relate to how we represent multiple qubits.

Tensor Products

The tensor (Kronecker) product of two matrices $A$ and $B$ is denoted $A \otimes B$.

Example: $$ \begin{bmatrix}a & b \\ c & d\end{bmatrix} \otimes \begin{bmatrix}e & f \\ g & h\end{bmatrix} = \begin{bmatrix} ae & af & be & bf \\ ag & ah & bg & bh \\ ce & cf & de & df \\ cg & ch & dg & dh \end{bmatrix} $$

Tensor products are used to build larger spaces from smaller components. In quantum computing, they describe how multi-part systems combine.

Practice

1) Compute $(2,3) + (4,1)$.

$(6,4)$

2) Find the dot product of $(1,2,3)$ and $(4,5,6)$.

$1*4 + 2*5 + 3*6 = 32$

3) For matrix $A = \begin{bmatrix}2 & 0 \\ 0 & 3\end{bmatrix}$, what are its eigenvalues?

$2$ and $3$

4) Compute the tensor product of $(1,0)$ and $(0,1)$.

$(0,1,0,0)$

Tensors

Vectors

Operations

Matrices

Operations

Eigenvalues and Eigenvectors

Tensors

Tensor Products

Practice

1) Compute \((2,3) + (4,1)\).

2) Find the dot product of \((1,2,3)\) and \((4,5,6)\).

3) For matrix \(A = \begin{bmatrix}2 & 0 \\ 0 & 3\end{bmatrix}\), what are its eigenvalues?

4) Compute the tensor product of \((1,0)\) and \((0,1)\).