Complex Numbers Have A Representation As Two By Two Matrices With Real Entries In This Problem You Will Be Asked To Sho 1 (37.06 KiB) Viewed 28 times
Complex numbers have a representation as two by two matrices with real entries. In this problem you will be asked to show that the matrix operation indeed reproduces the complex number operations. For any complex number z assign a matrix M₂ as follows: z = a + ib→ Mz For example we have M₁+2i = (a) Show that M21 + Mz2 and real number c. = = 1 -2 or M3i 2 1 Mz1+z2 and cMz = a 0 -3 3 0 Mez, for any complex numbers 2, 21, 22
(b) Show that Mz₁ Mz2 = Mz₁22, for any complex numbers 21, 22. 2122
(c) Show that M₁/2 = M₂¹, for any nonzero complex number z.
(d) Find an expression for det(M₂) and MT in terms of operations of complex numbers.
(e) Find the eigenvalues and corresponding eigenvectors of M₂ for z = a + ib.
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