Identity Matrix (original) (raw)

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The identity matrix is a the simplest nontrivial diagonal matrix, defined such that

I(X)=X (1)

for all vectors X. An identity matrix may be denoted 1, I,E (the latter being an abbreviation for the German term "Einheitsmatrix"; Courant and Hilbert 1989, p. 7), or occasionally I, with a subscript sometimes used to indicate the dimension of the matrix. Identity matrices are sometimes also known as unit matrices (Akivis and Goldberg 1972, p. 71).

The n×n identity matrix is given explicitly by

I_(ij)=delta_(ij) (2)

for i,j=1,2, ..., n, where delta_(ij) is the Kronecker delta. Written explicitly,

![ I=[1 0 ... 0; 0 1 ... 0; | ... ; 0 0 ... 1]. ](http://mathworld.wolfram.com/images/equations/IdentityMatrix/NumberedEquation3.svg) (3)

The n×n identity matrix is implemented in the Wolfram Language as IdentityMatrix[_n_].

"Square root of identity" matrices can be defined for I_n by solving

| [a_(11) a_(12) ... a_(1n); a_(21) a_(22) ... a_(2n); \| ... ... | ; a_(n1) a_(n2) ... a_(nn)][a_(11) a_(12) ... a_(1n); a_(21) a_(22) ... a_(2n); | ... ... | ; a_(n1) a_(n2) ... a_(nn)]=[1 0 ... 0; 0 1 ... 0; | | ... 0; 0 0 ... 1]. | (4) | | ------------------------------------------------------------------ | ------------------------------------------------------------------------------- | ------- | -------------------------------------------------- | | -------------------------------------------------------------------------------------------------------- | --- |

For n=2, the most general form of the resulting square root matrix is

I_2^(1/2)=[+/-d (1-d^2)/c; c ∓d],[+/-d c; (1-d^2)/c ∓d] (5)

giving

[+/-1 0; 0 +/-1],[+/-1 0; c ∓1],[+/-1 c; 0 ∓1] (6)

as limiting cases.

"Cube root of identity" matrices can take on even more complicated forms. However, one simple class of such matrices is called _k_-matrices.

See also

(0,1)-Matrix, Constant Matrix, Diagonal Matrix, _k_-Matrix,Scalar Matrix, Unit Matrix,Zero Matrix

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References

Akivis, M. A. and Goldberg, V. V. An Introduction to Linear Algebra and Tensors. New York: Dover, 1972.Ayres, F. Jr. Schaum's Outline of Theory and Problems of Matrices. New York: Schaum, p. 10, 1962.Courant, R. and Hilbert, D. Methods of Mathematical Physics, Vol. 1. New York: Wiley, 1989.

Referenced on Wolfram|Alpha

Identity Matrix

Cite this as:

Weisstein, Eric W. "Identity Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IdentityMatrix.html

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