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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,

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(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:

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

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

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

image5.png

Note that the preceding equations also mean the following:

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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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