diff --git a/spec/API_specification/linear_algebra_functions.md b/spec/API_specification/linear_algebra_functions.md
index 100efccf2..7cf85d29c 100644
--- a/spec/API_specification/linear_algebra_functions.md
+++ b/spec/API_specification/linear_algebra_functions.md
@@ -103,14 +103,59 @@ Returns the specified diagonals. If `x` has more than two dimensions, then the a
 TODO
 
 (function-eig)=
-### eig()
+### eig(x, /)
 
-TODO
+Computes the eigenvalues and eigenvectors of a square matrix (or a stack of square matrices) `x`.
 
-(function-eigvalsh)=
-### eigvalsh()
+#### Parameters
 
-TODO
+-   **x**: _<array>_
+
+    -   input array having shape `(..., M, M)` and whose innermost two dimensions form square matrices. Must have a floating-point data type.
+
+        TODO: support complex numbers
+
+#### Returns
+
+-   **out**: _Tuple\[ <array> ]_
+
+    -   a namedtuple (`e`, `v`) whose
+    
+        -   first element must have shape `(..., M)` and consist of computed eigenvalues
+        -   second element must have shape `(..., M, M)`and have the columns of the inner most matrices contain the computed eigenvectors.
+
+        TODO: add note regarding return types
+
+```{note}
+
+Eigenvalue sort order is left unspecified.
+```
+
+(function-eigvals)=
+### eigvals(x, /)
+
+Computes the eigenvalues of a square matrix (or a stack of square matrices) `x`.
+
+#### Parameters
+
+-   **x**: _<array>_
+
+    -   input array having shape `(..., M, M)` and whose innermost two dimensions form square matrices. Must have a floating-point data type.
+
+        TODO: support complex numbers
+
+#### Returns
+
+-   **out**: _<array>_
+
+    -   an array containing the computed eigenvalues. The array must have shape `(..., M)`.
+
+        TODO: add note regarding return data type.
+
+```{note}
+
+Eigenvalue sort order is left unspecified.
+```
 
 (function-einsum)=
 ### einsum()