/TensorFlow 2.4

tf.linalg.matmul

Multiplies matrix `a` by matrix `b`, producing `a` * `b`.

The inputs must, following any transpositions, be tensors of rank >= 2 where the inner 2 dimensions specify valid matrix multiplication dimensions, and any further outer dimensions specify matching batch size.

Both matrices must be of the same type. The supported types are: `float16`, `float32`, `float64`, `int32`, `complex64`, `complex128`.

Either matrix can be transposed or adjointed (conjugated and transposed) on the fly by setting one of the corresponding flag to `True`. These are `False` by default.

If one or both of the matrices contain a lot of zeros, a more efficient multiplication algorithm can be used by setting the corresponding `a_is_sparse` or `b_is_sparse` flag to `True`. These are `False` by default. This optimization is only available for plain matrices (rank-2 tensors) with datatypes `bfloat16` or `float32`.

A simple 2-D tensor matrix multiplication:

```a = tf.constant([1, 2, 3, 4, 5, 6], shape=[2, 3])
a  # 2-D tensor
<tf.Tensor: shape=(2, 3), dtype=int32, numpy=
array([[1, 2, 3],
[4, 5, 6]], dtype=int32)>
b = tf.constant([7, 8, 9, 10, 11, 12], shape=[3, 2])
b  # 2-D tensor
<tf.Tensor: shape=(3, 2), dtype=int32, numpy=
array([[ 7,  8],
[ 9, 10],
[11, 12]], dtype=int32)>
c = tf.matmul(a, b)
c  # `a` * `b`
<tf.Tensor: shape=(2, 2), dtype=int32, numpy=
array([[ 58,  64],
[139, 154]], dtype=int32)>
```

A batch matrix multiplication with batch shape [2]:

```a = tf.constant(np.arange(1, 13, dtype=np.int32), shape=[2, 2, 3])
a  # 3-D tensor
<tf.Tensor: shape=(2, 2, 3), dtype=int32, numpy=
array([[[ 1,  2,  3],
[ 4,  5,  6]],
[[ 7,  8,  9],
[10, 11, 12]]], dtype=int32)>
b = tf.constant(np.arange(13, 25, dtype=np.int32), shape=[2, 3, 2])
b  # 3-D tensor
<tf.Tensor: shape=(2, 3, 2), dtype=int32, numpy=
array([[[13, 14],
[15, 16],
[17, 18]],
[[19, 20],
[21, 22],
[23, 24]]], dtype=int32)>
c = tf.matmul(a, b)
c  # `a` * `b`
<tf.Tensor: shape=(2, 2, 2), dtype=int32, numpy=
array([[[ 94, 100],
[229, 244]],
[[508, 532],
[697, 730]]], dtype=int32)>
```

Since python >= 3.5 the @ operator is supported (see PEP 465). In TensorFlow, it simply calls the `tf.matmul()` function, so the following lines are equivalent:

```d = a @ b @ [[10], [11]]
d = tf.matmul(tf.matmul(a, b), [[10], [11]])
```
Args
`a` `tf.Tensor` of type `float16`, `float32`, `float64`, `int32`, `complex64`, `complex128` and rank > 1.
`b` `tf.Tensor` with same type and rank as `a`.
`transpose_a` If `True`, `a` is transposed before multiplication.
`transpose_b` If `True`, `b` is transposed before multiplication.
`adjoint_a` If `True`, `a` is conjugated and transposed before multiplication.
`adjoint_b` If `True`, `b` is conjugated and transposed before multiplication.
`a_is_sparse` If `True`, `a` is treated as a sparse matrix. Notice, this does not support `tf.sparse.SparseTensor`, it just makes optimizations that assume most values in `a` are zero. See `tf.sparse.sparse_dense_matmul` for some support for `tf.sparse.SparseTensor` multiplication.
`b_is_sparse` If `True`, `b` is treated as a sparse matrix. Notice, this does not support `tf.sparse.SparseTensor`, it just makes optimizations that assume most values in `a` are zero. See `tf.sparse.sparse_dense_matmul` for some support for `tf.sparse.SparseTensor` multiplication.
`name` Name for the operation (optional).
Returns
A `tf.Tensor` of the same type as `a` and `b` where each inner-most matrix is the product of the corresponding matrices in `a` and `b`, e.g. if all transpose or adjoint attributes are `False`:

`output[..., i, j] = sum_k (a[..., i, k] * b[..., k, j])`, for all indices `i`, `j`.

`Note` This is matrix product, not element-wise product.
Raises
`ValueError` If `transpose_a` and `adjoint_a`, or `transpose_b` and `adjoint_b` are both set to `True`.