diff --git a/include/cotila/matrix/math.h b/include/cotila/matrix/math.h
index 2322960..003f788 100644
--- a/include/cotila/matrix/math.h
+++ b/include/cotila/matrix/math.h
@@ -68,7 +68,7 @@ constexpr matrix<detail::remove_complex_t<T>, N, M> abs(const matrix<T, N, M> &v
 }
 
 /** @brief computes the sum of elements of a matrix
- *  @param m an \f$ M \times M \f$ matrix of type T
+ *  @param m an \f$ N \times M \f$ matrix of type T
  *  @return a scalar \f$ \sum\limits_{i} m_i \f$ of type T
  *
  *  Computes the sum of the elements of a matrix.
@@ -78,7 +78,7 @@ template <typename T, std::size_t N, std::size_t M> constexpr T sum(const matrix
 }
 
 /** @brief computes the minimum valued element
- *  @param m an \f$ M \times M \f$ matrix of type T
+ *  @param m an \f$ N \times M \f$ matrix of type T
  *  @return a scalar \f$ v_i \f$ of type T where \f$ v_i \leq v_j,\ \forall j \f$
  *
  *  Computes the minimum valued element of a matrix.
@@ -88,7 +88,7 @@ template <typename T, std::size_t N, std::size_t M> constexpr T min(const matrix
 }
 
 /** @brief computes the maximum valued element
- *  @param m an \f$ M \times M \f$ matrix of type T
+ *  @param m an \f$ N \times M \f$ matrix of type T
  *  @return a scalar \f$ v_i \f$ of type T where \f$ v_i \geq v_j,\ \forall j \f$
  *
  *  Computes the maximum valued element of a matrix.
@@ -326,7 +326,7 @@ constexpr T trace(const matrix<T, M, M> &m) {
 }
 
 /** @brief computes the elementwise square root
- *  @param m an \f$ M \times M \f$ matrix of type T
+ *  @param m an \f$ N \times M \f$ matrix of type T
  *  @return an \f$ M \times N \f$ matrix of type T, the element wise
  *  square root of \f$ \textbf{m} \f$
  *  Computes the elementwise square root of a matrix.
diff --git a/include/cotila/matrix/operators.h b/include/cotila/matrix/operators.h
index 325121b..bca22ac 100644
--- a/include/cotila/matrix/operators.h
+++ b/include/cotila/matrix/operators.h
@@ -70,7 +70,7 @@ constexpr matrix<T, N, M> operator+(T a, const matrix<T, N, M> &m) {
  *  @param b an \f$ N \times M \f$ matrix of type T
  *  @return \f$ \textbf{a} + \textbf{b} \f$ such that \f$ \left(\textbf{a} + \textbf{b}\right)_{ij} = \textbf{a}_{ij} + \textbf{b}_{ij} \f$
  *
- *  Computes the vector sum.
+ *  Computes the matrix sum.
  */
 template <typename T, std::size_t N, std::size_t M>
 constexpr matrix<T, N, M> operator+(const matrix<T, N, M> &a,
@@ -83,7 +83,7 @@ constexpr matrix<T, N, M> operator+(const matrix<T, N, M> &a,
  *  @param a a scalar of type T
  *  @return \f$ \textbf{m}a \f$ such that \f$ \left(\textbf{m} a\right)_{ij} = \textbf{m}_{ij} a \f$
  *
- *  Computes the sum of a matrix and a scalar.
+ *  Computes the product of a matrix and a scalar.
  */
 template <typename T, std::size_t N, std::size_t M>
 constexpr matrix<T, N, M> operator*(const matrix<T, N, M> &m, T a) {
@@ -95,7 +95,7 @@ constexpr matrix<T, N, M> operator*(const matrix<T, N, M> &m, T a) {
  *  @param m an \f$ N \times M \f$ matrix of type T
  *  @return \f$ a\textbf{m} \f$ such that \f$ \left(a\textbf{m}\right)_{ij} = a\textbf{m}_{ij} \f$
  *
- *  Computes the sum of a matrix and a scalar.
+ *  Computes the product of a matrix and a scalar.
  */
 template <typename T, std::size_t N, std::size_t M>
 constexpr matrix<T, N, M> operator*(T a, const matrix<T, N, M> &m) {
@@ -107,7 +107,7 @@ constexpr matrix<T, N, M> operator*(T a, const matrix<T, N, M> &m) {
  *  @param b an \f$ N \times M \f$ matrix of type T
  *  @return \f$ \textbf{a} \circ \textbf{b} \f$ such that \f$ \left(\textbf{a} \circ \textbf{b}\right)_{ij} = \textbf{a}_{ij} \textbf{b}_{ij} \f$
  *
- *  Computes the Hadamard, or elementwise, product of two vectors.
+ *  Computes the Hadamard, or elementwise, product of two matrices.
  */
 template <typename T, std::size_t N, std::size_t M>
 constexpr matrix<T, N, M> operator*(const matrix<T, N, M> &a,
diff --git a/include/cotila/matrix/utility.h b/include/cotila/matrix/utility.h
index f4abe04..96e651c 100644
--- a/include/cotila/matrix/utility.h
+++ b/include/cotila/matrix/utility.h
@@ -13,7 +13,7 @@ namespace cotila {
 
 
 /** @brief accumulates an operation across a matrix
- *  @param m an \f$ M \times M \f$ matrix of type T
+ *  @param m an \f$ N \times M \f$ matrix of type T
  *  @param init the initial value
  *  @param f a function of type F that operates between U and matrix elements of type T
  *  @return \f$ f\left(f\left(\ldots f\left(\textrm{init}, \textbf{v}_1\right), \ldots\right), \textbf{v}_N \right) \f$
@@ -119,10 +119,10 @@ constexpr matrix<T, N, M> fill(T value) {
 }
 
 /** @brief shifts matrix elements
- *  @param m an \f$ M \times M \f$ matrix of type T
+ *  @param m an \f$ N \times M \f$ matrix of type T
  *  @param n the amount to shift each element
- *  @return an N-vector of type T \f$ \textbf{v} \gg n \f$ such that
- *  \f$ \left(\textbf{v} \gg n\right)_i = \textbf{v}_{(i + n)\ \textrm{mod}\ N} \f$
+ *  @return an \f$ N \times M \f$ matrix of type T such that
+ *  \f$ \left(\textbf{m} \gg n\right)_{ij} = \textbf{v}_{kj} with k = (i + n)\ \textrm{mod}\ N \f$
  *
  *  Rotates a matrix by shifting its elements vertically.
  */
diff --git a/include/cotila/vector/vector.h b/include/cotila/vector/vector.h
index 3c62277..4dca372 100644
--- a/include/cotila/vector/vector.h
+++ b/include/cotila/vector/vector.h
@@ -19,6 +19,13 @@ struct matrix;
 template <typename T, std::size_t N, typename Container>
 constexpr matrix<T, N, 1> from_initializer(Container &);
 
+/** @brief A container representing a vector
+ *  @tparam T scalar type to contain
+ *  @tparam N number of rows
+ *
+ *  `cotila::vector` is a container representing a vector.
+ *  It inherits all its properties from `̀cotila::matrix`.
+ */
 template <typename T, std::size_t N>
 struct vector : public matrix<T, N, 1> {
   constexpr vector() = default;