Section 28

Examples of linear transformations of Euclidean space. Adjoint transformation

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Subsection 28.1Examples of linear transformations of Euclidean space (28.1-28.9)[9/100%]
Problem 28.1

Let a1,…,aka_{1}, \ldots , a_{k} be an orthonormal basis in the subspace L⊂E\mathcal{L} \subset \mathcal{E}. The coordinate columns of these vectors in an orthonormal basis ee of the space E\mathcal{E} form the matrix AA. Write, in the basis ee, the matrix of

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(1)

orthogonal projection onto the space L\mathcal{L}

(2)

the matrix of reflection in the subspace L\mathcal{L}.

Problem 28.2

Let a1,…,aka_{1}, \ldots , a_{k} be a basis in the subspace L⊂E\mathcal{L} \subset \mathcal{E}. The coordinate columns of these vectors in an orthonormal basis ee of the space E\mathcal{E} form the matrix AA. Write, in the basis ee, the matrix:

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(1)

of orthogonal projection onto the subspace L\mathcal{L}

(2)

the matrix of reflection in the subspace L\mathcal{L}.

Problem 28.3

Let a1,…,aka_{1}, \ldots , a_{k} be a basis in the subspace L⊂E\mathcal{L} \subset \mathcal{E}. The coordinate columns of these vectors in a basis ee of the space E\mathcal{E} form the matrix AA. The Gram matrix Γ\Gamma of the basis e\mathbf{e} is given. Write, in the basis ee, the matrix:

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(1)

of orthogonal projection onto the subspace L\mathcal{L}

(2)

the matrix of reflection in the subspace L\mathcal{L}.

Problem 28.4
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A vector aa is given, and the subspace L⊂E\mathcal{L} \subset \mathcal{E} is defined by the equation (a,x)=0(a, x) = 0. Find the image of the vector xx under reflection in L\mathcal{L} and express the matrix of this transformation in terms of the coordinate column a of the vector aa in an orthonormal basis.

(2)

Let a vector xx and a vector yy of length 1 be given. Find an (n−1)(n - 1)-dimensional subspace, under reflection in which xx is carried into the vector λy(λ>0)\lambda y\quad (\lambda > 0). What is λ\lambda equal to?

Problem 28.5

Find the matrix of reflection in an (n−1)(n - 1)-dimensional subspace that carries the vector xx into the vector yy. The vectors are given by their coordinate columns ξ\xi and η\eta in an orthonormal basis:

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ξ=[12],  η=[21]\xi =\left[\begin{smallmatrix} 1 \\ 2\end{smallmatrix}\right], \; \eta =\left[\begin{smallmatrix} 2 \\ 1\end{smallmatrix}\right]

(2)

ξ=[−12],  η=[1−2]\xi =\left[\begin{smallmatrix} -1 \\ 2\end{smallmatrix}\right], \; \eta =\left[\begin{smallmatrix} 1 \\ -2\end{smallmatrix}\right]

(3)

ξ=[111],  η=[1−11]\xi =\left[\begin{smallmatrix} 1 \\ 1 \\ 1\end{smallmatrix}\right], \; \eta =\left[\begin{smallmatrix} 1 \\ - 1 \\ 1\end{smallmatrix}\right]

(4)

ξ=[21−2],  η=[030]\xi =\left[\begin{smallmatrix} 2 \\ 1 \\ - 2\end{smallmatrix}\right], \; \eta =\left[\begin{smallmatrix} 0 \\ 3 \\ 0\end{smallmatrix}\right]

(5)

ξ=[1021],  η=[2−110]\xi =\left[\begin{smallmatrix} 1 \\ 0 \\ 2 \\ 1\end{smallmatrix}\right], \; \eta =\left[\begin{smallmatrix} 2 \\ - 1 \\ 1 \\ 0\end{smallmatrix}\right].

Problem 28.6

Determine the geometric meaning of the transformation given in an orthonormal basis by the matrix:

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(1)

(1/2)A468(1 / 2) \mathbf{A}_{468}. A468=[111−111−111−111−1111]\mathbf{A}_{468} = \left[\begin{smallmatrix} 1 & 1 & 1 & -1 \\ 1 & 1 & -1 & 1 \\ 1 & -1 & 1 & 1 \\ -1 & 1 & 1 & 1 \end{smallmatrix}\right].

(2)

(1/3)A202(1 / 3) \mathbf{A}_{202}. A202=[12221−22−21]\mathbf{A}_{202} = \left[\begin{smallmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ 2 & -2 & 1 \end{smallmatrix}\right].

(3)

(1/18)A350(1 / 18) \mathbf{A}_{350}. A350=[1−41−416−41−41]\mathbf{A}_{350} = \left[\begin{smallmatrix} 1 & -4 & 1 \\ -4 & 16 & -4 \\ 1 & -4 & 1 \end{smallmatrix}\right].

(4)

A430\mathbf{A}_{430}. A430=[000−100−1001001000]\mathbf{A}_{430} = \left[\begin{smallmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{smallmatrix}\right].

(5)

A431\mathbf{A}_{431}. A431=[00−10000110000−100]\mathbf{A}_{431} = \left[\begin{smallmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{smallmatrix}\right].

Problem 28.7

Consider a four-dimensional Euclidean space and an orthonormal basis ee in it. In all cases the rotation is carried out in the direction from the first of the indicated vectors to the second. Write the matrix:

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(1)

of a rotation by π/2\pi / 2 in the linear span L\mathcal{L} of the vectors e2+e3e_{2} + e_{3} and e1−e4e_{1} - e_{4} (the vectors of L⊥\mathcal{L}^{\perp } are fixed);

(2)

of a rotation by π/4\pi / 4 in the linear span L\mathcal{L} of the vectors e1+e4e_{1} + e_{4} and e2−e3e_{2} - e_{3} (the vectors of L⊥\mathcal{L}^{\perp } are fixed);

(3)

of a rotation by π/6\pi / 6 in the linear span L\mathcal{L} of the vectors e1−e2e_{1} - e_{2} and e3−e4e_{3} - e_{4} (the vectors of L⊥\mathcal{L}^{\perp } are fixed);

(4)

of a rotation by π/4\pi / 4 in the linear span L\mathcal{L} of the vectors e1+e2e_{1} + e_{2} and e3+e4e_{3} + e_{4}, and by π/4\pi / 4 in the linear span L\mathcal{L} of the vectors e3−e4e_{3} - e_{4} and e1−e2e_{1} - e_{2}.

Problem 28.8

Let f1,…,fmf_{1}, \ldots , f_{m} and g1,…,gmg_{1}, \ldots , g_{m} be two systems of vectors in an nn-dimensional Euclidean space. Their coordinate columns in an orthonormal basis form, respectively, the matrices FF and GG. Find the matrix of the transformation

φ(x)=∑j=1m(x,fj)gj. \varphi (x) = \sum _{j = 1}^{m}\left(x, f_{j}\right) g_{j}.
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Problem 28.9

In the subspaces A,B⊂E\mathcal{A}, \mathcal{B} \subset \mathcal{E}, orthonormal bases a1,…,aka_{1}, \ldots , a_{k} and b1,…,bmb_{1}, \ldots , b_{m} are chosen. A vector x∈Ax\in \mathcal{A} is projected onto B\mathcal{B}, and then the resulting projection is projected onto A\mathcal{A}. This defines a transformation of the subspace A\mathcal{A}. Find its matrix in the basis a1,…,aka_{1}, \ldots , a_{k}. Prove that the eigenvalues of this transformation belong to the interval [0,1]\left[0, 1\right].

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Subsection 28.2The conjugate transformation (28.10-28.37)[28/54%]
Problem 28.10

Let there be, for some transformation φ\varphi of a Euclidean space, a transformation φ∗\varphi^{ * } such that (φ(x),y)=(x,φ∗(y))(\varphi (x), y) = \left(x, \varphi^{ * }(y)\right) for any vectors xx and yy. Prove that both transformations are linear.

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Problem 28.11

Find the transformation adjoint to an arbitrary transformation φ\varphi of a one-dimensional Euclidean space.

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Problem 28.12

Find the transformation adjoint to the transformation of problem 28. 8.

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Problem 28.13

Prove that two transformations commute if and only if their adjoint transformations commute.

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Problem 28.14

Let the transformation φ\varphi be nilpotent. Prove that its adjoint transformation is also nilpotent with the same index of nilpotency.

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Problem 28.15

Prove that for two transformations φ\varphi and ψ\psi the product φ∗ψ=o\varphi^{ * } \psi = o if and only if Im⁡φ\operatorname {Im} \varphi is orthogonal to Im⁡ψ\operatorname {Im} \psi.

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Problem 28.16

Let φ\varphi be a rotation of the plane by an angle α\alpha. Find the adjoint transformation φ∗\varphi^{ * }.

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Problem 28.17

Let aˉ\bar{a} be a fixed vector of three-dimensional geometric space. The transformation φ\varphi assigns to each vector xˉ\bar{x} the cross product [aˉ,xˉ]\left[\bar{a}, \bar{x}\right]. Find φ∗\varphi^{ * }.

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Problem 28.18

In an arithmetic space with the standard scalar product, a transformation φ\varphi carries the vectors a1,a2,a3a_{1}, a_{2}, a_{3} respectively into the vectors b1,b2,b3b_{1}, b_{2}, b_{3}. Find the matrix of the adjoint transformation φ∗\varphi^{ * }:

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(1)

a1=c99,a2=c52,a3=c51;b1=c110,b2=c89,b3=c64a_{1} = \mathbf{c}_{99}, a_{2} = \mathbf{c}_{52}, a_{3} = \mathbf{c}_{51}; b_{1} = \mathbf{c}_{110}, b_{2} = \mathbf{c}_{89}, b_{3} = \mathbf{c}_{64}. c99=[−111],c52=[1−11],c51=[11−1],c110=[2−11],c89=[12−1],c64=[−112]\mathbf{c}_{99} = \left[\begin{smallmatrix} -1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{52} = \left[\begin{smallmatrix} 1 \\ -1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{51} = \left[\begin{smallmatrix} 1 \\ 1 \\ -1 \end{smallmatrix}\right], \mathbf{c}_{110} = \left[\begin{smallmatrix} 2 \\ -1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{89} = \left[\begin{smallmatrix} 1 \\ 2 \\ -1 \end{smallmatrix}\right], \mathbf{c}_{64} = \left[\begin{smallmatrix} -1 \\ 1 \\ 2 \end{smallmatrix}\right].

(2)

a1=c142,a2=c66,a3=c144;b1=c83,b2=c139,b3=c104a_{1} = \mathbf{c}_{142}, a_{2} = \mathbf{c}_{66}, \quad a_{3} = \mathbf{c}_{144}; \quad b_{1} = \mathbf{c}_{83}, b_{2} = \mathbf{c}_{139}, b_{3} = \mathbf{c}_{104}. c142=[011],c66=[111],c144=[110],c83=[123],c139=[−2−2−2],c104=[1−23]\mathbf{c}_{142} = \left[\begin{smallmatrix} 0 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{66} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{144} = \left[\begin{smallmatrix} 1 \\ 1 \\ 0 \end{smallmatrix}\right], \mathbf{c}_{83} = \left[\begin{smallmatrix} 1 \\ 2 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{139} = \left[\begin{smallmatrix} -2 \\ -2 \\ -2 \end{smallmatrix}\right], \mathbf{c}_{104} = \left[\begin{smallmatrix} 1 \\ -2 \\ 3 \end{smallmatrix}\right].

(3)

a1=c57,a2=c77,a3=c64;b1=c145,b2=c101,b3=c67a_{1} = \mathbf{c}_{57}, a_{2} = \mathbf{c}_{77}, a_{3} = \mathbf{c}_{64}; b_{1} = \mathbf{c}_{145}, b_{2} = \mathbf{c}_{101}, b_{3} = \mathbf{c}_{67}. c57=[100],c77=[0−11],c64=[−112],c145=[20−1],c101=[1−21],c67=[012]\mathbf{c}_{57} = \left[\begin{smallmatrix} 1 \\ 0 \\ 0 \end{smallmatrix}\right], \mathbf{c}_{77} = \left[\begin{smallmatrix} 0 \\ -1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{64} = \left[\begin{smallmatrix} -1 \\ 1 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{145} = \left[\begin{smallmatrix} 2 \\ 0 \\ -1 \end{smallmatrix}\right], \mathbf{c}_{101} = \left[\begin{smallmatrix} 1 \\ -2 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{67} = \left[\begin{smallmatrix} 0 \\ 1 \\ 2 \end{smallmatrix}\right].

Problem 28.19

The matrix AA of a transformation φ\varphi in a basis ee with Gram matrix Γ\Gamma is given. Find the matrix of the adjoint transformation φ∗\varphi^{ * }:

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(1)

A=A50,Γ=A19A = \mathbf{A}_{50}, \quad \Gamma = \mathbf{A}_{19}. A50=[0−11−1],A19=[1112]\mathbf{A}_{50} = \left[\begin{smallmatrix} 0 & -1 \\ 1 & -1 \end{smallmatrix}\right], \mathbf{A}_{19} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 2 \end{smallmatrix}\right].

(2)

A=A16,Γ=A104A = \mathbf{A}_{16}, \quad \Gamma = \mathbf{A}_{104}. A16=[1−111],A104=[5−3−32]\mathbf{A}_{16} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & 1 \end{smallmatrix}\right], \mathbf{A}_{104} = \left[\begin{smallmatrix} 5 & -3 \\ -3 & 2 \end{smallmatrix}\right].

(3)

A=A20,Γ=A104A = \mathbf{A}_{20}, \quad \Gamma = \mathbf{A}_{104}. A20=[0−110],A104=[5−3−32]\mathbf{A}_{20} = \left[\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right], \mathbf{A}_{104} = \left[\begin{smallmatrix} 5 & -3 \\ -3 & 2 \end{smallmatrix}\right].

(4)

A=A260,Γ=A178A = \mathbf{A}_{260}, \quad \Gamma = \mathbf{A}_{178}. A260=[010001100],A178=[641452121]\mathbf{A}_{260} = \left[\begin{smallmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{smallmatrix}\right], \mathbf{A}_{178} = \left[\begin{smallmatrix} 6 & 4 & 1 \\ 4 & 5 & 2 \\ 1 & 2 & 1 \end{smallmatrix}\right].

(5)

A=A199,Γ=A176A = \mathbf{A}_{199}, \quad \Gamma = \mathbf{A}_{176}. A199=[010000000],A176=[101010102]\mathbf{A}_{199} = \left[\begin{smallmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{smallmatrix}\right], \mathbf{A}_{176} = \left[\begin{smallmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 2 \end{smallmatrix}\right].

(6)

A=A261,Γ=A176A = \mathbf{A}_{261}, \quad \Gamma = \mathbf{A}_{176}. A261=[103212301],A176=[101010102]\mathbf{A}_{261} = \left[\begin{smallmatrix} 1 & 0 & 3 \\ 2 & 1 & 2 \\ 3 & 0 & 1 \end{smallmatrix}\right], \mathbf{A}_{176} = \left[\begin{smallmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 2 \end{smallmatrix}\right].

(7)

A=A259,Γ=A177A = \mathbf{A}_{259}, \quad \Gamma = \mathbf{A}_{177}. A259=[001100010],A177=[110121012]\mathbf{A}_{259} = \left[\begin{smallmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{smallmatrix}\right], \mathbf{A}_{177} = \left[\begin{smallmatrix} 1 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2 \end{smallmatrix}\right].

(8)

A=A260,Γ=A179A = \mathbf{A}_{260}, \quad \Gamma = \mathbf{A}_{179}. A260=[010001100],A179=[3−21−22−11−11]\mathbf{A}_{260} = \left[\begin{smallmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{smallmatrix}\right], \mathbf{A}_{179} = \left[\begin{smallmatrix} 3 & -2 & 1 \\ -2 & 2 & -1 \\ 1 & -1 & 1 \end{smallmatrix}\right].

(9)

A=A269,Γ=A388A = \mathbf{A}_{269}, \quad \Gamma = \mathbf{A}_{388}. A269=[023010002],A388=[210121011]\mathbf{A}_{269} = \left[\begin{smallmatrix} 0 & 2 & 3 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{smallmatrix}\right], \mathbf{A}_{388} = \left[\begin{smallmatrix} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 1 \end{smallmatrix}\right].

Problem 28.20

In a three-dimensional arithmetic space with the standard scalar product, find the transformation adjoint to the projection onto the line x=2y=3zx = 2y = 3z parallel to the plane z=0z = 0.

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Problem 28.21

In the space P2\mathcal{P}^{2} of polynomials of degree at most 2 with the standard scalar product, a transformation δ\delta assigns to a polynomial its derivative. Find the adjoint transformation δ∗\delta^{ * }. Write the matrix of δ∗\delta^{ * }:

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(1)

in the basis 1,t,t21, t, t^{2},

(2)

in the basis 1,t,(3t2−1)/21, t, \left(3t^{2} - 1\right) / 2.

Problem 28.22

In the space P2\mathcal{P}^{2} of polynomials of degree at most 2 with the scalar product defined in problem 25.8,325. 8, 3, a transformation δ\delta assigns to a polynomial its derivative. Find the adjoint transformation δ∗\delta^{ * }. Write the matrix of δ∗\delta^{ * }:

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in the basis 1,t,t21, t, t^{2},

(2)

in the basis 1,t,(3t2−1)/21, t, \left(3t^{2} - 1\right) / 2.

Problem 28.23

Let AA be a square matrix of order nn. To any square matrix XX of the same order the matrix φ(X)=AX\varphi (X) = AX is assigned. This defines a linear transformation of the space of square matrices with the standard scalar product. Find its adjoint transformation φ∗\varphi^{ * }.

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Problem 28.24

Let AA be a nondegenerate square matrix of order nn. To any square matrix XX of the same order the matrix φ(X)=A−1XA\varphi (X) = A^{ - 1} XA is assigned. This defines a linear transformation of the space of square matrices with the standard scalar product. Find its adjoint transformation φ∗\varphi^{ * }.

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Problem 28.25

Prove the following properties of the operation of passing to the adjoint transformation in a Euclidean space:

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(1)

(φ+ψ)∗=φ∗+ψ∗(\varphi + \psi )^{ * } = \varphi^{ * } + \psi^{ * }

(2)

(φψ)∗=ψ∗φ∗(\varphi \psi )^{ * } = \psi^{ * } \varphi^{ * }

(3)

(αφ)∗=αφ∗(\alpha \varphi )^{ * } = \alpha \varphi^{ * }

(4)

If φ\varphi has an inverse, then φ∗\varphi^{ * } is also invertible, and (φ∗)−1=(φ−1)∗\left(\varphi^{ * }\right)^{ - 1} = \left(\varphi^{ - 1}\right)^{ * }.

Problem 28.26

Prove that for mutually adjoint transformations of a Euclidean space the following coincide:

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the ranks,

(2)

the characteristic polynomials,

(3)

the eigenvalues,

(4)

the dimensions of the eigenspaces.

Problem 28.27

Let the transformation φ\varphi be diagonalizable. Prove that φ∗\varphi^{ * } is also diagonalizable.

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Problem 28.28

Let ee be a basis of eigenvectors of the transformation φ\varphi. Prove that its biorthogonal basis e∗\mathbf{e}^{ * } consists of eigenvectors of the adjoint transformation φ∗\varphi^{ * }.

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Problem 28.29

Let the transformation φ\varphi have matrix AA in the basis ee. Prove that φ∗\varphi^{ * } has matrix ATA^{T} in the biorthogonal basis e∗\mathbf{e}^{ * }.

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Problem 28.30
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Prove that the set of values of a linear transformation φ\varphi coincides with the orthogonal complement of the kernel of the adjoint transformation φ∗\varphi^{ * }.

(2)

Verify that statement 1) is equivalent to the Fredholm theorem for systems of linear equations.

Problem 28.31

Find the set of values of the transformation δ∗\delta^{ * } from problem 28. 21 and directly verify that it coincides with the orthogonal complement of the kernel of φ\varphi.

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Problem 28.32

Find the kernel of the transformation φ\varphi from problem 28. 17 and directly verify that it coincides with the orthogonal complement of the set of values of φ∗\varphi^{ * }.

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Problem 28.33
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Let λ1\lambda_{1} and λ2\lambda_{2} be distinct eigenvalues of a transformation φ\varphi of a Euclidean space. Prove that Ker⁡(φ−λ1ι)⊆Im⁡(φ−λ2ι)\operatorname {Ker}\left(\varphi - \lambda_{1} \iota \right) \subseteq \operatorname {Im}\left(\varphi - \lambda_{2} \iota \right)

(2)

Is such a statement true for a transformation of a vector space?

Problem 28.34
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Let the subspace L⊂E\mathcal{L} \subset \mathcal{E} be invariant under the transformation φ\varphi. Prove that L⊥\mathcal{L}^{\perp } is invariant under φ∗\varphi^{ * }.

(2)

Let the transformation φ\varphi have a real characteristic number. Prove that it has an (n−1)(n - 1)-dimensional invariant subspace. Is the converse true?

(3)

Let all roots of the characteristic polynomial of the transformation φ\varphi be real. Prove that there exists an orthonormal basis in which the matrix of the transformation is upper triangular.

Problem 28.35

A transformation φ\varphi is given in an orthonormal basis by the matrix AA. Find an orthonormal basis in which its matrix A′A^{\prime } is upper triangular, and write the matrix A′A^{\prime }:

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(1)

A=[31−1022−1−11]A =\left[\begin{smallmatrix} 3 & 1 & - 1 \\ 0 & 2 & 2 \\ - 1 & - 1 & 1\end{smallmatrix}\right]

(2)

A=A221A = \mathbf{A}_{221}. A221=[41−125−244−1]\mathbf{A}_{221} = \left[\begin{smallmatrix} 4 & 1 & -1 \\ 2 & 5 & -2 \\ 4 & 4 & -1 \end{smallmatrix}\right].

(3)

A=A292A = \mathbf{A}_{292}. A292=[1−2−1−11110−1]\mathbf{A}_{292} = \left[\begin{smallmatrix} 1 & -2 & -1 \\ -1 & 1 & 1 \\ 1 & 0 & -1 \end{smallmatrix}\right].

(4)

A=A290A = \mathbf{A}_{290}. A290=[21−33−2−311−2]\mathbf{A}_{290} = \left[\begin{smallmatrix} 2 & 1 & -3 \\ 3 & -2 & -3 \\ 1 & 1 & -2 \end{smallmatrix}\right].

(5)

A=A261A = \mathbf{A}_{261}. A261=[103212301]\mathbf{A}_{261} = \left[\begin{smallmatrix} 1 & 0 & 3 \\ 2 & 1 & 2 \\ 3 & 0 & 1 \end{smallmatrix}\right].

Problem 28.36

Let φ\varphi be a linear transformation of a Euclidean space. Prove that the root subspaces of φ\varphi and φ∗\varphi^{ * } belonging to unequal eigenvalues are orthogonal.

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Problem 28.37

Let φ\varphi be a linear transformation of a Euclidean space. How are the Jordan forms of the transformations φ\varphi and φ∗\varphi^{ * } related?

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