# Tensors and matrices¶

Tensors are global variables provided by Taichi. Tensors can be either sparse or dense. An element of a tensor can be either a scalar or a vector/matrix.

Although mathematically matrices are treated as 2D tensors, in Taichi, tensor and matrix are two completely different things. Matrices can be used as tensor elements, so you have tensors of matrices.

## Tensors of scalars¶

• Every global variable is an N-dimensional tensor.
• Global scalars are treated as 0-D tensors of scalars.
• Tensors are accessed using indices, e.g. `x[i, j, k]` if `x` is a scalar 3D tensor. For a 0-D tensor, access it as `x[None]`.
• Even when accessing 0-D tensor `x`, use `x[None] = 0` instead of `x = 0`. Please always use indexing to access entries in tensors.
• Tensor values are initially zero.
• Sparse tensors are initially inactive.
• See Tensors of scalars for more details.

## Tensors of matrices¶

Suppose you have a `128 x 64` global grid `A`, each node containing a `3 x 2` matrices. In this case you need to allocate a `128 x 64` tensor of `3 x 2` matrix, using the statement `A = ti.Matrix(3, 2, dt=ti.f32, shape=(128, 64))`.

• If you want to get the matrix of grid node `i, j`, please use `mat = A[i, j]`. `mat` is simply a `3 x 2` matrix
• To get the element on the first row and second column of that matrix, use `mat[0, 1]` or `A[i, j][0, 1]`.
• As you may have noticed, there are two indexing operators `[]`, the first is for tensor indexing, the second for matrix indexing.
• For a tensor `F` of element `ti.Matrix`, make sure you first index the tensor dimensions, and then the matrix dimensions: `F[i, j, k][0, 2]`. (Assuming `F` is a 3D tensor with `ti.Matrix` of size `3x3` as elements)
• `ti.Vector` is simply an alias of `ti.Matrix`.
• See Matrices for more on matrices.

## Matrix size¶

For performance reasons matrix operations will be unrolled, therefore we suggest using only small matrices. For example, `2x1`, `3x3`, `4x4` matrices are fine, yet `32x6` is probably too big as a matrix size.

Warning

Due to the unrolling mechanisms, operating on large matrices (e.g. `32x128`) can lead to long compilation time and low performance.

If you have a dimension that is too large (e.g. `64`), it’s better to declare a tensor of size `64`. E.g., instead of declaring `ti.Matrix(64, 32, dt=ti.f32, shape=(3, 2))`, declare `ti.Matrix(3, 2, dt=ti.f32, shape=(64, 32))`. Try to put large dimensions to tensors instead of matrices.