- #1
Hall
- 351
- 88
- Homework Statement
- The same as the title.
- Relevant Equations
- A matrix is said to be unitary if A*A = AA* = I. A
A matrix ##A## is diagonalisable if there is a matrix C such that
##\Lambda = C^{-1} A C ##.
Let's assume that ##A## is unitary and diagonalisable, so, we have
## \Lambda = C^{-1} A C ##
Since, ##\Lambda## is made up of eigenvalues of ##A##, which is unitary, we have ## \Lambda \Lambda^* = \Lambda \bar{\Lambda} = I##.
I tried using some, petty, algebra to prove that ##C C* = I## but was unsuccessful. Here is my attempt:
## \Lambda = C^{-1} A C ##
## \Lambda^* = C^* A^* (C^*)^{-1} ##
## \Lambda \Lambda^* = C ^{-1} A C ~ C^* A^* (C^{-1})^{*} ##
## I = C ^{-1} A C ~ C^* A^* (C^{-1})^{*} ##
## I = C^{-1} AC ~ [ C^{-1} A C ]^{*} ##
That implies,
## [ C^{-1} A C ]^{*} = [C^{-1} A C]^{-1} ##
## C^* A^* (C^{-1})^{*} = C^{-1} A^* C##, since ##A^* = A^{-1}##.
But I cannot conclude ##C^* =C^{-1}##.
## \Lambda = C^{-1} A C ##
Since, ##\Lambda## is made up of eigenvalues of ##A##, which is unitary, we have ## \Lambda \Lambda^* = \Lambda \bar{\Lambda} = I##.
I tried using some, petty, algebra to prove that ##C C* = I## but was unsuccessful. Here is my attempt:
## \Lambda = C^{-1} A C ##
## \Lambda^* = C^* A^* (C^*)^{-1} ##
## \Lambda \Lambda^* = C ^{-1} A C ~ C^* A^* (C^{-1})^{*} ##
## I = C ^{-1} A C ~ C^* A^* (C^{-1})^{*} ##
## I = C^{-1} AC ~ [ C^{-1} A C ]^{*} ##
That implies,
## [ C^{-1} A C ]^{*} = [C^{-1} A C]^{-1} ##
## C^* A^* (C^{-1})^{*} = C^{-1} A^* C##, since ##A^* = A^{-1}##.
But I cannot conclude ##C^* =C^{-1}##.