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Problem 7: Unitary Matrices: A complex, square matrix $U$ is called unitary if it satisfies $U^dagger U = I$, where $U^dagger = overline{U^T}$ denotes the Hermitian transpose in which one first transposes and then takes complex conjugates of all entries. (a) Show that $U$ is a unitary matrix if and only if $U^{-1} = U^dagger$. (b) Show that the following matrix is unitary and compute its inverse: $egin{pmatrix} frac{1}{sqrt{2}} & frac{i}{sqrt{2}} \ frac{i}{sqrt{2}} & frac{1}{sqrt{2}} end{pmatrix}$.

          Problem 7: Unitary Matrices: A complex, square matrix $U$ is called unitary if it satisfies $U^dagger U = I$, where $U^dagger = overline{U^T}$ denotes the Hermitian transpose in which one first transposes and then takes complex conjugates of all entries.
(a) Show that $U$ is a unitary matrix if and only if $U^{-1} = U^dagger$.
(b) Show that the following matrix is unitary and compute its inverse:
$egin{pmatrix} frac{1}{sqrt{2}} & frac{i}{sqrt{2}} \ frac{i}{sqrt{2}} & frac{1}{sqrt{2}} end{pmatrix}$.

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$Problem 7: Unitary Matrices: A complex, square matrix U is called unitary if it satisfies U^dagger U = I, where U^dagger = overlineU^T denotes the Hermitian transpose in which one first transposes and then takes complex conjugates of all entries. (a) Show that U is a unitary matrix if and only if U^-1 = U^dagger. (b) Show that the following matrix is unitary and compute its inverse: eginpmatrix frac1sqrt2 fracisqrt2 fracisqrt2 frac1sqrt2 endpmatrix.$

Added by Nicol-S H.

Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert Hsin-Chieh Liao

06/16/2022

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Problem: Unitary Matrices A matrix U is called unitary if it satisfies U*U = I, where U* denotes the Hermitian transpose, which involves transposing and taking complex conjugates of all entries. Show that a matrix U is unitary if and only if U*U = I. Show that the following matrix is unitary and compute its inverse:

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00:01 To show, first we want to show that u is a unitary matrix, if and only if u inverse equal to u star.

00:09 And you start just means you, this definition is just, this is just u bar and then take bar and then take transpose.

00:22 Or you can use also you transpose and take bar.

00:25 This is the same.

00:29 Yeah.

00:29 Second is to show this u equal to these.

00:32 This is an i, you turn it unitary, and compute it's inverse.

00:38 So first, let's do a.

00:43 And, okay, so by definition, so u is unitary, this just means that, okay, so u -star, u is equal to i, identity, metrics.

00:59 So what we can do is multiply both sides by u -inverse on the right.

01:05 So u star u -un -un -unverse is equal to iu -unverse.

01:14 So we get iu -inverse, it's just u -unverse.

01:18 And in this case, because this is u -star, u -u -inverse, so this becomes i, so this becomes u -star.

01:35 So this implies, okay, so u -star is equal to u -inverse.

01:40 So if you is unitary, we have you star equal to u inverse.

01:44 And conversely, if we have you, if you star is equal to u inverse, we want to see if u is unitary.

02:02 So we just check and try the definition.

02:09 So you start u, this is the same as u inverse in you because u star is u inverse and u inverse time you is i.

02:16 So u star u is equal to i.

02:18 Yeah, so, so yeah, so.

02:21 U is unitary.

02:26 So we are done for a and for b, we want to show these metrics is unitary.

02:47 We want to show these metrics is unitary...

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