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How to Find the Inverse of a Large Matrix

Finding the inverse of a large matrix often isn’t easy, so quantum physics calculations are sometimes limited to working with unitary operators, U, where the operator’s inverse is equal to its adjoint,

image0.png

(To find the adjoint of an operator, A, you find the transpose by interchanging the rows and columns, AT. Then take the complex conjugate,

image1.png

Note that the asterisk (*) symbol in the above equation means the complex conjugate. (A complex conjugate flips the sign connecting the real and imaginary parts of a complex number.)

This gives you the following equation:

image2.png

The product of two unitary operators, U and V, is also unitary because

image3.png

When you use unitary operators, kets and bras transform this way:

image4.png

And you can transform other operators using unitary operators like this:

image5.png

Note that the preceding equations also mean the following:

image6.png

Here are some properties of unitary transformations:

  • If an operator is Hermitian, then its unitary transformed version,

    image7.png
  • is also Hermitian.

  • The eigenvalues of A and its unitary transformed version,

    image8.png
  • are the same.

  • Commutators that are equal to complex numbers are unchanged by unitary transformations:

    image9.png
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