z=a+ib→Mz= a -b ; b a. (a) Show that Mz1 + Mz2 = Mz1+z2 and cMz = Mcz, for any complex numbers z, z1, z2 and real number c. (b) Show that Mz1xMz2 = Mz1z2 , for any complex numbers z1, z2. c) Show that M1/z = Mz^−1, for any nonzero complex number z. (d) Find an expression for det(Mz) and Mz^T in terms of operations of complex numbers. (e) Find the eigenvalues and corresponding eigenvectors of Mz for z = a + ib. (
z=a+ib→Mz= a -b ; b a. (a) Show that Mz1 + Mz2 = Mz1+z2 and cMz = Mcz, for any complex numbers z, z1, z2 and real number c. (b) Show that Mz1xMz2 = Mz1z2 , for any complex numbers z1, z2. c) Show that M1/z = Mz^−1, for any nonzero complex number z. (d) Find an expression for det(Mz) and Mz^T in terms of operations of complex numbers. (e) Find the eigenvalues and corresponding eigenvectors of Mz for z = a + ib. (
z=a+ib→Mz= a -b ; b a. (a) Show that Mz1 + Mz2 = Mz1+z2 and cMz = Mcz, for any complex numbers z, z1, z2 and real number c. (b) Show that Mz1xMz2 = Mz1z2 , for any complex numbers z1, z2. c) Show that M1/z = Mz^−1, for any nonzero complex number z. (d) Find an expression for det(Mz) and Mz^T in terms of operations of complex numbers. (e) Find the eigenvalues and corresponding eigenvectors of Mz for z = a + ib. (
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 Mz as follows:
z=a+ib→Mz= a -b ; b a.
(a) Show that Mz1 + Mz2 = Mz1+z2 and cMz = Mcz, for any complex numbers z, z1, z2 and real number c.
(b) Show that Mz1xMz2 = Mz1z2 , for any complex numbers z1, z2.
c) Show that M1/z = Mz^−1, for any nonzero complex number z.
(d) Find an expression for det(Mz) and Mz^T in terms of operations of complex numbers.
(e) Find the eigenvalues and corresponding eigenvectors of Mz for z = a + ib. (Note that you can only use the techniques used in the section 4.1).
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Transcribed Image Text:(c) Show that M₁/2 = M₂¹, for any nonzero complex number z.
2
(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. (Note that
you can only use the techniques used in the section 4.1).
Transcribed Image Text:z = a + ib→ M₂
For example we have M1+2i =
21
(a) Show that M₁ + Mz2
and real number c.
=
=
[1 -2
or M3i
2
1
Mz₁+z2 and cM₂
a
b
=
=
-b
3 0
Mez, for any complex numbers 2, 21, 22
(b) Show that M2₁ Mz2 = Mz1₁z2, for any complex numbers Z1, Z2.
22
Combination of a real number and an imaginary number. They are numbers of the form a + b , where a and b are real numbers and i is an imaginary unit. Complex numbers are an extended idea of one-dimensional number line to two-dimensional complex plane.
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