# Does this special identity matrix have a name?

recently I stumbled upon the problem of defining a diagonal matrix whose elements are identity matrices of $$dim = n$$, where $$n$$ is the column/row index. For example, for $$n=3$$:

$$\mathbb{I}_3 = \left[{\begin{array}{ccc} I_1 & 0 & 0 \\ 0 & I_2 & 0 \\ 0 & 0 & I_3 \end{array} }\right]$$,

and the subscript indicates the size of the matrix, i.e., $$I_2$$ is a $$2\times 2$$ identity matrix and so on.

This definition may look silly, but I need a matrix with this property to explicitly define the direct sum of matrices with a notation that's more usual than $$\bigoplus_i^n$$.

So, $$\mathbb{I}_n$$ does have a special name?

• Unless I am mistaken, your example is $I_6$. – Mindlack Jan 21 '19 at 17:07
• It looks like, but $\mathbb{I}_3$ is actually a 3x3 matrix, where the elements of the diagonal are progressively larger identity matrices. This shape is neccessary for the mathematical properties I need. – Eduardo W. Jan 21 '19 at 17:24
• You mean, it is not a block notation, but a matrix of which the entries are matrices? – Mindlack Jan 21 '19 at 17:26
• Precisely! The 1st element is a 1x1 identity matrix. The 2nd element on the diagonal is a 2x2 identity. The 3rd is a 3x3 identity and so on. So, expanding the elements back into full matrices, the full dimension of $\mathbb{I}_n$ is $n!$. – Eduardo W. Jan 21 '19 at 17:27
• I don't think what you're asking for has/needs a special name. Generally you talk about matrices built out of smaller matrices as being "block matrices" and then you use your arguments about how to compute with the blocks. – rschwieb Jan 21 '19 at 17:29

There’s not a special unique name, but in general, $$\mathbb{I}_{n} = I_{\frac{n(n+1)}{2}}$$.
In your example, $$n=3$$ so $$\dfrac{n(n+1)}{2} = 6$$