A review of inner product spaces and matrix algebra


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  • The following topics will be covered in this lecture:
    • An introduction to arrays in Python
    • Basic vector operations
    • Orthogonality
    • Subspaces
    • Orthogonal projection lemma
    • Gram-Schmidt and QR decomposition
    • Matrix / vector multiplication
    • Matrix / matrix multiplication
    • Special classes of matrices

A review of inner product spaces

  • Linear algebra is a fundamental concept for applying and understanding statistical methods in more than one variable.

    • This is at the basis of formulating multivariate distributions and random vectors, as well as their analysis.
  • More specifically in data assimilation, algorithm design is strongly shaped by numerical stability and scalability;

    • for this reason, an understanding of vector subspaces, projections and matrix factorizations is critical for performing dimensional / computational reductions.
  • We will not belabor the details and proofs of these results, as these can be found in other classes / books devoted to the subject.

    • Likewise, it will be assumed that in computation, optimized numerical linear algebra libraries like LAPACK and OpenBLAS (or their wrappers like numpy) will be utilized.
  • For this reason, these lectures will survey a variety of results from an applied perspective, providing intuition to how and why these tools are used.

  • We will start by introducing the basic characteristics of vectors / matrices, their operations and their implementation in Numpy.

  • Along the way, we will introduce some essential language and concepts about vector spaces, inner product spaces, linear transformations and important tools in applied matrix algebra.

Pythonic programming

  • Python uses several standard scientific libraries for numerical computing, data processing and visualization.
  • At the core, there is a Python kernel and interpreter that can take human readable inputs and turn these into machine code.
  • This is the basic Python functionality, but there are extensive specialized libraries.
  • The most important of these for scientific computing are the following:
    1. Numpy – designed for large array manipulation in vectorized operations;
    2. Scipy – a library of numerical routines and scientific computing ecosystem;
    3. Pandas – R Dataframe inspired, data structures and analysis;
    4. Scikit-learn – a general regression and machine learning library;
    5. Matplotlib – Matlab inspired, object oriented plotting and visualization library.

Numpy arrays

  • To accommodate the flexibility of the Python programming environment, conventions around methods name spaces and scope have been adopted.

    • The convention is to utilize import statements to call methods of the library.
    • For example, we will import the library numpy as a new object to call methods from
import numpy as np
  • The tools we use from numpy will now be called from numpy as an object, with the form of the call looking like np.method()

  • Numpy has a method known as “array”;

my_vector = np.array([1,2,3])
array([1, 2, 3])
  • Notice that we can identify properties of an array, such as its dimensions, as follows:

Numpy arrays continued

  • Arrays are the object class in numpy that handles both vector and matrix objects:
my_array = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
array([[1, 2, 3],
       [4, 5, 6],
       [7, 8, 9]])
(3, 3)

Numpy arrays continued

  • Note that numpy arrays function as mathematical multi-linear matricies in arbitrary dimensions:
my_3D_array = np.array([[[1, 2], [3, 4]], [[5, 6], [7, 8]]])
array([[[1, 2],
        [3, 4]],

       [[5, 6],
        [7, 8]]])
(2, 2, 2)

Array notations

  • Mathematically, we will define the vector notations \( \pmb{x} \in \mathbb{R}^{N_x} \), matrix notations \( \mathbf{A} \in \mathbb{R}^{N_x \times N_x} \), and matrix-slice notations \( \mathbf{A}^j \in \mathbb{R}^{N_x} \) as

    \[ \begin{align} \pmb{x} := \begin{pmatrix} x_1 \\ \vdots \\ x_{N_x} \end{pmatrix} & & \mathbf{A} := \begin{pmatrix} a_{1,1} & \cdots & a_{1, N_x} \\ \vdots & \ddots & \vdots \\ a_{N_x,1} & \cdots & a_{N_x, N_x} \end{pmatrix} & & \mathbf{A}^j := \begin{pmatrix} a_{1,j} \\ \vdots \\ a_{N_x, j} \end{pmatrix} \end{align} \]

  • Elements of the matrix \( \mathbf{A} \) may further be referred to by index in row and column as

    \[ \mathbf{A}_{i,j} = \mathbf{A}\left[i,j\right] = a_{i,j} \]

  • In numpy, we can make a reference to sub-arrays analogously with the : slice notation:

array([[1, 2, 3],
       [4, 5, 6]])
array([1, 4, 7])

Array operations

  • Because arrays are understood as mathematical objects, they have inherent methods for mathematical computation.

  • Suppose we have two vectors

    \[ \begin{align} \pmb{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}\in\mathbb{R}^{3 \times 1} & & \pmb{y} = \begin{pmatrix} y_1 \\ y_2 \\ y_3 \end{pmatrix}\in \mathbb{R}^{3\times 1} \end{align} \]

  • We can perform basic mathematical operations on these element-wise as follows

    \[ \begin{align} \pmb{x} + \pmb{y} = \begin{pmatrix} x_1 + y_1 \\ x_2 + y_2 \\ x_3 + y_3 \end{pmatrix} & & \pmb{x}\circ\pmb{y} = \begin{pmatrix} x_1 * y_1 \\ x_2 * y_2 \\ x_3 * y_3 \end{pmatrix} \end{align} \]

  • Both of these operations generalize to vectors of arbitrary length.

Numpy arrays continued

  • In Python the syntax for such operations is given by
x = np.array([1, 2, 3])
y = np.array([4, 5, 6])
array([5, 7, 9])
array([ 4, 10, 18])
  • The simple, element-wise multiplication and addition of vectors can be performed on any arrays of matching dimension as above.

  • This type of multiplication is known as the Schur product of arrays.

The Schur product of arrays
Let \( \mathbf{A},\mathbf{B} \in \mathbb{R}^{N \times M} \) be arrays of arbitrary dimension with \( N,M \geq 1 \). The Schur product is defined \[ \begin{align} \mathbf{A}\circ \mathbf{B}:= \begin{pmatrix}a_{1,1} * b_{1,1} & \cdots & a_{1,M}* b_{1,M} \\ \vdots & \ddots & \vdots\\ a_{N,1}* b_{N,1} & \cdots & a_{N,M} * b_{N,M} \end{pmatrix} \end{align} \]

Euclidean inner product

  • The array-valued Schur product is not the most widely-used array product;

    • rather, the scalar-valued inner product and its extension to general matrix multiplication will be more common.
  • Notice that the two previously defined vectors \( \pmb{x} \) and \( \pmb{y} \) were defined as column vectors

    \[ \begin{align} \pmb{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}\in\mathbb{R}^{3 \times 1} & & \pmb{y} = \begin{pmatrix} y_1 \\ y_2 \\ y_3 \end{pmatrix}\in \mathbb{R}^{3\times 1} \end{align} \]

  • The transpose of \( \pmb{x} \) is defined as the row vector,

    \[ \begin{align} \pmb{x}^\top = \begin{pmatrix} x_1 & x_2 & x_3 \end{pmatrix} \in \mathbb{R}^{1 \times 3} \end{align} \]

  • The standard, Euclidean vector inner product is defined for the vectors \( \pmb{x} \) and \( \pmb{y} \) as follows

    \[ \begin{align} \pmb{x}^\top \pmb{y} = x_1 * y_1 + x_2 * y_2 + x_3 * y_3 \end{align} \]

  • That is, we take each row element from \( \pmb{x}^\top \) and multiply it by each column element of \( \pmb{y} \) and take the sum of these products.

  • This generalizes to vectors of arbitrary length \( N_x \) as,

    \[ \begin{align} \pmb{x}^\top \pmb{y} = \sum_{i=1}^{N_x} x_i * y_i \end{align} \]

The Euclidean norm and inner product

The Euclidean inner product
For two vectors \( \pmb{x},\pmb{y} \in\mathbb{R}^{N_x} \), the Euclidean inner product is given as \[ \begin{align} \langle \pmb{x}, \pmb{y}\rangle := \pmb{x}^\top \pmb{y} = \sum_{i=1}^{N_x} \pmb{x}_i *\pmb{y}_i \end{align} \]
  • The previous formula arises by formally extending the Euclidean distance formula to arbitrary dimensions.
Euclidean norm
Let \( \pmb{x}\in\mathbb{R}^{N_x} \). The Euclidean norm of \( \pmb{x} \) is defined \[ \begin{align} \parallel \pmb{x}\parallel := \sqrt{ \sum_{i=1}^{N_x} x_i^2} \equiv \sqrt{\pmb{x}^\top\pmb{x}} \end{align} \]
  • It is important to note that there are other distances that can be defined on \( \mathbb{R}^{N_x} \) different than the Euclidean distance;

    • in particular, the Euclidean distance represents a “flat” distance in all directions without any preference or penalty.
  • Note, it can be shown that the Euclidean inner product satisfies

    \[ \begin{align} \pmb{x}^\top\pmb{y} = \parallel \pmb{x} \parallel * \parallel \pmb{y} \parallel \cos\left(\theta\right), \end{align} \] where,

    1. \( \parallel \pmb{x}\parallel \) refers to the Euclidean length of the vector, defined as \( \parallel \pmb{x}\parallel =\sqrt{\pmb{x}^\top\pmb{x}} \); and
    2. \( \theta \) is the angle formed by the two vectors \( \pmb{x} \) and \( \pmb{y} \) at the origin \( \boldsymbol{0} \).

The Euclidean norm and inner product

  • Following our previous example, we will demonstrate the array transpose function and the dot product:

  • Recall that x had the following dimensions

  • If we compare x and its transpose, we see
array([1, 2, 3])
array([1, 2, 3])
  • This is due to the fact that numpy does not distinguish between row and column vectors.

    • The transpose() function extends, however, to two-dimensional arrays in the usual fashion.

The Euclidean norm and inner product

  • We can therefore compute the “dot” or Euclidean inner product several different ways:
x @ y
  • The @ notation refers to general matrix multiplication, which we will discuss shortly.


  • Notice that the equation

    \[ \begin{align} \pmb{x}^\top\pmb{y} = \parallel \pmb{x} \parallel * \parallel \pmb{y} \parallel \cos\left(\theta\right), \end{align} \]

    generalizes the idea of a perpendicular angle between lines;

    • the above product is zero if and only if \( \theta = \frac{\pi}{2} + k *\pi \) for any \( k\in \mathbb{Z} \).
Orthogonal vectors
We say that two vectors \( \pmb{x},\pmb{y} \) are orthogonal if and only if \[ \begin{align} \pmb{x}^\top \pmb{y} = \pmb{0} \end{align} \]
  • Notice that with scalar / vector multiplication defined as

    \[ \tilde{\pmb{x}}:= \alpha * \pmb{x}:= \begin{pmatrix} \alpha * x_1 \\ \vdots \\ \alpha * x_{N_x}\end{pmatrix} \]


    \[ \begin{align} \tilde{\pmb{x}}^\top \pmb{y} = 0 & & \Leftrightarrow & & \pmb{x}^\top \pmb{y} = 0 \end{align} \]

  • This brings us to an important notions of linear combinations and subspaces.

Linear combinations and subspaces

  • The scalar multiples of \( \pmb{x} \) give a simple example of linear combinations of vectors.
Linear combination
Let \( n\geq 1 \) be an arbitrary integer, \( \alpha_i\in\mathbb{R} \) and \( \pmb{x}_i\in\mathbb{R}^{N_x} \) for each \( i=1,\cdots,n \). Then \[ \begin{align} \pmb{x} := \sum_{i=1}^n \alpha_i \pmb{x}_i \end{align} \] is a linear combination of the vectors \( \{\pmb{x}_i\}_{i=1}^{n} \).
  • A subspace can then be defined from linear combinations of vectors as follows.
The collection of vectors \( V\subset \mathbb{R}^{N_x} \) is denoted a subspace if and only if for any arbitrary collection vectors \( \pmb{x}_i\in V \) and scalars \( \alpha_i\in\mathbb{R} \), their linear combination \[ \sum_{i=1}^n \alpha_i \pmb{x}_i = \pmb{x} \in V. \]
  • With the above linear combinations in mind, we will use the following notation

    \[ \begin{align} \mathrm{span}\{\pmb{x}_i\}_{i=1}^n := \left\{\pmb{x}\in\mathbb{R}^{N_x} : \exists\text{ } \alpha_i \text{ for which } \pmb{x}= \sum_{i=1}^{n}\alpha_i \pmb{x}_i\right\} . \end{align} \]

  • It can be readily seen then that the span of any collection of vectors is a subspace by construction.

Linear independence and bases

  • Related notions are linear independence, dependence and bases
Linear independence / dependence
Let \( \pmb{x}\in\mathbb{R}^{N_x} \) and \( \pmb{x}_i\in\mathbb{R}^{N_x} \) for \( i=1,\cdots,n \). The vector \( \pmb{x} \) is linearly independent (respectively dependent) with the collection \( \{\pmb{x}_i\}_{i=1}^{n} \) if and only if \( \pmb{x}\notin \mathrm{span}\{\pmb{x}_i\}_{i=1}^{n} \) (respectively \( \pmb{x}\in \mathrm{span}\{\pmb{x}_i\}_{i=1}^{n} \)).
  • It is clear then that, e.g., \( \pmb{x}_1 \) is linearly dependent with \( \mathrm{span}\{\pmb{x}_i\}_{i=1}^n \) trivially.

  • A related idea is whether for some vector \( \pmb{x}\in \mathrm{Span}\{\pmb{x}_i\}_{i=1}^n \) the choice of the scalar coefficients \( \alpha_i \) defining \( \pmb{x}=\sum_{i=1}^n \alpha_x \pmb{x}_i \) is unique.

Let \( V\subset \mathbb{R}^{N_x} \) be a subspace. A collection \( \{\pmb{x}_i\}_{i=1}^{n} \) is said to be a basis for \( V \) if \( V = \mathrm{span}\{\pmb{x}_i\}_{i=1}^n \) and if \[ \begin{align} \pmb{0} = \sum_{i=1}^n \alpha_i \pmb{x}_i \end{align} \] holds if and only if \( \alpha_i=0 \) \( \forall i \).
  • In particular, a choice of a basis for \( V \) gives a unique coordinatization of any vector \( \pmb{x}\in V \).

    • If we suppose there existed two coordinatizatons for a vector \( \pmb{x} \) in the basis \( \{\pmb{x}_i\}_{i=1}^n \),

    \[ \begin{align} \pmb{x}=\sum_{i=1}^n \alpha_i \pmb{x}_i = \sum_{i=1}^n \beta_i \pmb{x}_i & & \Leftrightarrow & &\pmb{0} = \sum_{i=1}^n \left(\alpha_i - \beta_i \right) \pmb{x}_i \end{align} \] and all \( \beta_i = \alpha_i \).

Orthogonal bases and subspaces

  • When we define a choice of inner product, such as the Euclidean inner product, a special class of basis is often useful for theoretical / computational purposes.
Orthogonal (Orthonormal) bases
Let \( \{\pmb{x}_i\}_{i=1}^n \) define a basis for \( V \subset \mathbb{R}^{N_x} \). The basis is said to be orthogonal if and only if each pair of basis vectors is orthogonal. A basis is said to orthonormal if, moreover, each basis vector has norm equal to one. In particular, for an orthonormal basis, if \[ \begin{align} \pmb{x} = \sum_{i=1}^n \alpha_i \pmb{x}_i \end{align} \] then \( \alpha_i = \pmb{x}_i^\top \pmb{x} \).
  • The above property shows that we can recover the “projection” coefficient \( \alpha_i \) of \( \pmb{x} \) into \( V \) using the inner product of the vector \( \pmb{x} \) with the basis vector \( \pmb{x}_i \).

    • This is a critical property, which we will generalize after we define orthogonal subspaces.
Orthogonal subspaces
The subspaces \( W,V\subset \mathbb{R}^{N_x} \) are orthogonal if and only if for every \( \pmb{y}\in W \) and \( \pmb{x}\in V \), \[ \begin{align} \pmb{y}^\top \pmb{x} = \pmb{0}. \end{align} \] Orthogonal subspaces will be denoted with \( W \perp V \).
  • With these constructions in mind, we can now introduce two of the most fundamental tools of inner product spaces:

    • orthogonal projections; and
    • the Gram-Schmidt process.

Orthogonal projections

  • When we think of orthogonal projections, we can think about the way the shadow of an object is projected onto two dimensions via the sun.
    • Particularly, the orthogonal projection would correspond to high-noon with the sun directly overhead.