The Identity Matrix

The matrix version of the number 1: a square matrix with ones down the main diagonal and zeros elsewhere, which leaves any matrix unchanged under multiplication in either order.

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In ordinary multiplication the number 1 leaves everything unchanged. Matrices have their own version of that number, and it is called the identity matrix.

Concept The identity element

For ordinary numbers, multiplying by 1 changes nothing:

The identity matrix plays exactly this role for matrix multiplication. It is written , with a subscript for its size when that matters: , , and so on.

Concept What it looks like
100 010 001 the main diagonal runs top-left to bottom-right
Every entry on the main diagonal is 1; every other entry is 0. The matrix is always square, since only a square matrix has a full diagonal.

There is one identity matrix for every size — , , and so on upward.

Theorem The identity property

Multiplying in either order returns the original matrix — one of the rare cases where matrix multiplication does not care about order. The sizes must match: an matrix needs .

Example Checking it entry by entry

Multiply by .

Entry (1,1):
Entry (1,2):
Entry (2,1):
Entry (2,2):
⟹ the same matrix A

The zeros switch off every term except one, and the single 1 lets that term through untouched. That is the whole mechanism.

Summary
  1. The identity element for numbers is 1; for matrices it is the identity matrix I.
  2. I is square, with 1s on the main diagonal and 0s everywhere else.
  3. There is one for each size: I₂, I₃, I₄ and so on.
  4. A × I = I × A = A, in either order.
  5. The two matrices must be the same size for the multiplication to work.
283. Relations and Functions
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