diff --git a/spec/API_specification/linear_algebra_functions.md b/spec/API_specification/linear_algebra_functions.md
index 209a2f787..f1a5ee60c 100644
--- a/spec/API_specification/linear_algebra_functions.md
+++ b/spec/API_specification/linear_algebra_functions.md
@@ -144,10 +144,31 @@ TODO
 
 TODO
 
-(function-matrix_power)=
-### matrix_power()
+(function-linalg-matrix_power)=
+### linalg.matrix_power(x, n, /)
 
-TODO
+Raises a square matrix (or a stack of square matrices) `x` to an integer power `n`.
+
+#### Parameters
+
+-   **x**: _<array>_
+
+    -   input array having shape `(..., M, M)` and whose innermost two dimensions form square matrices. Should have a floating-point data type.
+
+-   **n**: _int_
+
+    -   integer exponent.
+
+#### Returns
+
+-   **out**: _<array>_
+
+    -   if `n` is equal to zero, an array containing the identity matrix for each square matrix. If `n` is less than zero, an array containing the inverse of each square matrix raised to the absolute value of `n`, provided that each square matrix is invertible. If `n` is greater than zero, an array containing the result of raising each square matrix to the power `n`. The returned array must have the same shape as `x` and a floating-point data type determined by {ref}`type-promotion`.
+
+#### Raises
+
+-   if the innermost two dimensions of `x` are not the same size (i.e., form square matrices).
+-   if `n` is less than zero and a square matrix is not invertible.
 
 (function-matrix_rank)=
 ### matrix_rank()