fix small typos

Author: HumphreyYang

lectures/eigen_I.md

@@ -190,7 +190,7 @@ One way to understand this transformation is that $A$
 
 Let's examine some standard transformations we can perform with matrices.
 
-Below we visualise transformations by thinking of vectors as points
+Below we visualize transformations by thinking of vectors as points
 instead of arrows.
 
 We consider how a given matrix transforms 
@@ -511,7 +511,7 @@ Let $A$ be the $90^{\circ}$ clockwise rotation matrix given by
 $\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$ and let $B$ be a shear matrix
 along the x-axis given by $\begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$.
 
-We will visualise how a grid of points changes when we apply the
+We will visualize how a grid of points changes when we apply the
 transformation $AB$ and then compare it with the transformation $BA$.
 
 ```{code-cell} ipython3
@@ -685,7 +685,7 @@ In this case, repeatedly multiplying a vector by $A$ makes the vector "spiral ou
 
 We thus observe that the sequence $(A^kv)_{k \geq 0}$ behaves differently depending on the map $A$ itself.
 
-We now discuss the property of A that determines this behaviour.
+We now discuss the property of A that determines this behavior.
 
 (la_eigenvalues)=
 ## Eigenvalues 
@@ -846,7 +846,7 @@ This is discussed further later.
 ```{exercise}
 :label: eig1_ex1
 
-Power iteration is a method for finding the largest absolute eigenvalue of a diagnalizable matrix.
+Power iteration is a method for finding the largest absolute eigenvalue of a diagonalizable matrix.
 
 The method starts with a random vector $b_0$ and repeatedly applies the matrix $A$ to it
 
@@ -1087,17 +1087,17 @@ for i, example in enumerate(examples):
 plt.show()
 ```
 
-The vector fields explains why we observed the trajectories of the vector $v$ multiplied by $A$ iteratively before.
+The vector fields explain why we observed the trajectories of the vector $v$ multiplied by $A$ iteratively before.
 
 The pattern demonstrated here is because we have complex eigenvalues and eigenvectors.
 
 It is important to acknowledge that there is a complex plane.
 
-If we add the complex axis for the plot, the plot will be more complicated.
+If we add the complex axis to the plot, the plot will be more complicated.
 
 Here we used the real part of the eigenvalues and eigenvectors.
 
-We can try to plot the complex plane for one of the matrix using `Arrow3D` class retrieved from [stackoverflow](https://stackoverflow.com/questions/22867620/putting-arrowheads-on-vectors-in-matplotlibs-3d-plot).
+We can try to plot the complex plane for one of the matrices using `Arrow3D` class retrieved from [stackoverflow](https://stackoverflow.com/questions/22867620/putting-arrowheads-on-vectors-in-matplotlibs-3d-plot).
 
 ```{code-cell} ipython3
 class Arrow3D(FancyArrowPatch):

lectures/eigen_II.md

@@ -73,15 +73,15 @@ Here are some examples to illustrate this further.
 
 Let $A$ be a square nonnegative matrix and let $A^k$ be the $k^{th}$ power of $A$.
 
-A matrix is consisdered **primitive** if there exists a $k \in \mathbb{N}$ such that $A^k$ is everywhere positive.
+A matrix is considered **primitive** if there exists a $k \in \mathbb{N}$ such that $A^k$ is everywhere positive.
 
 It means that $A$ is called primitive if there is an integer $k \geq 0$ such that $a^{k}_{ij} > 0$ for *all* $(i,j)$.
 
 We can see that if a matrix is primitive, then it implies the matrix is irreducible.
 
-This is becuase if there exists an $A^k$ such that $a^{k}_{ij} > 0$ for all $(i,j)$, then it guarantees the same property for ${k+1}^th, {k+2}^th ... {k+n}^th$ iterations.
+This is because if there exists an $A^k$ such that $a^{k}_{ij} > 0$ for all $(i,j)$, then it guarantees the same property for ${k+1}^th, {k+2}^th ... {k+n}^th$ iterations.
 
-In other words, a primitive matrix is both irreducible and aperiodical as aperiodicity requires the a state to be visited with a guarantee of returning to itself after certain amount of iterations.
+In other words, a primitive matrix is both irreducible and aperiodical as aperiodicity requires a state to be visited with a guarantee of returning to itself after a certain amount of iterations.
 
 ### Left Eigenvectors
 
@@ -97,7 +97,7 @@ A vector $\varepsilon$ is called a left eigenvector of $A$ if $\varepsilon$ is a
 
 In other words, if $\varepsilon$ is a left eigenvector of matrix A, then $A^T \varepsilon = \lambda \varepsilon$, where $\lambda$ is the eigenvalue associated with the left eigenvector $v$.
 
-This hints on how to compute left eigenvectors
+This hints at how to compute left eigenvectors
 
 ```{code-cell} ipython3
 # Define a sample matrix
@@ -134,10 +134,10 @@ This is a more common expression and where the name left eigenvectors originates
 (perron-frobe)=
 ### The Perron-Frobenius Theorem
 
-For a nonnegative matrix $A$ the behaviour of $A^k$ as $k \to \infty$ is controlled by the eigenvalue with the largest
+For a nonnegative matrix $A$ the behavior of $A^k$ as $k \to \infty$ is controlled by the eigenvalue with the largest
 absolute value, often called the **dominant eigenvalue**.
 
-For a matrix $A$, the Perron-Frobenius theorem characterises certain
+For a matrix $A$, the Perron-Frobenius theorem characterizes certain
 properties of the dominant eigenvalue and its corresponding eigenvector when
 $A$ is a nonnegative square matrix.
 
@@ -214,7 +214,7 @@ Using matrix algebra we can conclude that the solution to this system of equatio
 What guarantees the existence of a unique vector $x^{*}$ that satisfies
 {eq}`neumann_eqn` ?
 
-The following is a fundamental result in functional analysis that generalises
+The following is a fundamental result in functional analysis that generalizes
 {eq}`gp_sum` to a multivariate case.
 
 (neumann_series_lemma)=

lectures/markov_chains_I.md

@@ -1085,7 +1085,7 @@ P :=
 \right)
 $$
 
-where rows, from top to down, correspondes to growth, stagnation and collapse.
+where rows, from top to down, correspond to growth, stagnation, and collapse.
 
 In this exercise,
 

lectures/markov_chains_II.md

@@ -30,7 +30,7 @@ In addition to what's in Anaconda, this lecture will need the following librarie
 
 This lecture continues the journey in Markov chain.
 
-Specifically, we will introduce irreducibility and ergodicity, and how they connect to stionarity.
+Specifically, we will introduce irreducibility and ergodicity, and how they connect to stationarity.
 
 Irreducibility is a concept that describes the ability of a Markov chain to move between any two states in the system.