diff --git a/spec/API_specification/linear_algebra_functions.md b/spec/API_specification/linear_algebra_functions.md
index 209a2f787..3b99eb3f2 100644
--- a/spec/API_specification/linear_algebra_functions.md
+++ b/spec/API_specification/linear_algebra_functions.md
@@ -149,10 +149,26 @@ TODO
 
 TODO
 
-(function-matrix_rank)=
-### matrix_rank()
+(function-linalg-matrix_rank)=
+### linalg.matrix_rank(x, /, *, rtol=None)
 
-TODO
+Computes the rank (i.e., number of non-zero singular values) of a matrix (or a stack of matrices).
+
+#### Parameters
+
+-   **x**: _<array>_
+
+    -   input array having shape `(..., M, N)` and whose innermost two dimensions form `MxN` matrices. Should have a floating-point data type.
+
+-   **rtol**: _Optional\[ Union\[ float, <array> ] ]_
+
+    -   relative tolerance for small singular values. Singular values less than or equal to `rtol * largest_singular_value` are set to zero. If a `float`, the value is equivalent to a zero-dimensional array having a floating-point data type determined by {ref}`type-promotion` (as applied to `x`) and must be broadcast against each matrix. If an `array`, must have a floating-point data type and must be compatible with `shape(x)[:-2]` (see {ref}`broadcasting`). If `None`, the default value is `max(M, N) * eps`, where `eps` must be the machine epsilon associated with the floating-point data type determined by {ref}`type-promotion` (as applied to `x`). Default: `None`.
+
+#### Returns
+
+-   **out**: _<array>_
+
+    -   an array containing the ranks. The returned array must have a floating-point data type determined by {ref}`type-promotion` and must have shape `(...)` (i.e., must have a shape equal to `shape(x)[:-2]`).
 
 (function-norm)=
 ### norm(x, /, *, axis=None, keepdims=False, ord=None)