grandes-ecoles

QI.A.1 Groups Symplectic and Orthogonal Group Properties View

Show that $A \in \mathrm{SO}(2)$ if and only if there exists a real $t$ such that $A = R_t$ with $R_t = \left(\begin{array}{rr} \cos t & -\sin t \\ \sin t & \cos t \end{array}\right)$.

QI.A.2 Groups Symplectic and Orthogonal Group Properties View

Write a procedure or function in Maple or Mathematica that takes as input a quadruple $(a,b,c,d)$ of reals and returns, when possible, a real $t$ such that $\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) = R_t$ and an error message otherwise.

QI.A.3 Groups Group Homomorphisms and Isomorphisms View

Verify that the map which associates to every real $t$ the matrix $R_t$ is a surjective homomorphism from the group $(\mathbb{R},+)$ onto the group $(\mathrm{SO}(2),\times)$. Is this homomorphism bijective?

QI.A.4 Groups Group Homomorphisms and Isomorphisms View

Show that, for all $t$ in $\mathbb{R}$ and all non-zero $u$ in $\mathbb{R}^2$, $t$ is a measure of the oriented angle $(u\widehat{\rho_t(u)})$, where $\rho_t$ is the endomorphism (the rotation of angle $t$) $f_{R_t}$ canonically associated with $R_t$.

QI.B.1 Invariant lines and eigenvalues and vectors Properties of eigenvalues under matrix operations View

Let $A, B \in \mathcal{M}_n(\mathbb{R})$ and $P \in \mathrm{GL}_n(\mathbb{R})$ such that $B = P^{-1}AP$. Show that $f_A$ and $f_B$ have the same eigenvalues.

QI.B.2 Invariant lines and eigenvalues and vectors Compute eigenvectors or eigenspaces View

Let $A, B \in \mathcal{M}_n(\mathbb{R})$ and $P \in \mathrm{GL}_n(\mathbb{R})$ such that $B = P^{-1}AP$. Show that if $\lambda$ is an eigenvalue of $A$, then $E_\lambda(f_A) = f_P(E_\lambda(f_B))$.

QI.C.1 Invariant lines and eigenvalues and vectors Geometric interpretation of eigenstructure View

We denote $K_2 = \left(\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right)$. Verify that the endomorphism $\sigma_0 = f_{K_2}$ is a reflection (orthogonal symmetry with respect to a line in the plane) and specify its eigenelements.

QI.C.2 Invariant lines and eigenvalues and vectors Geometric interpretation of eigenstructure View

For all real $t$, specify the endomorphism $\sigma_t$ canonically associated with $R_t^{-1} K_2 R_t$ and in particular its eigenelements.

QI.C.3 Groups Symplectic and Orthogonal Group Properties View

Show that for every matrix $A$ of $\mathrm{O}(2)$ such that $\det(A) = -1$, there exists a real $t$ such that $$A = \left(\begin{array}{cr} \cos(2t) & \sin(2t) \\ \sin(2t) & -\cos(2t) \end{array}\right)$$

QII.A.1 Groups Group Homomorphisms and Isomorphisms View

Show that for all $A$ in $\mathcal{M}_n(\mathbb{R})$ we have $A$ dos $A$, that for all $(A,B)$ in $\mathcal{M}_n(\mathbb{R})^2$ if $A$ dos $B$ then $B$ dos $A$, and that for all $(A,B,C)$ in $\mathcal{M}_n(\mathbb{R})^3$ if $A$ dos $B$ and $B$ dos $C$ then $A$ dos $C$.

QII.A.2 Groups Group Homomorphisms and Isomorphisms View

What are the matrices directly orthogonally similar to $\alpha I_n$ for $\alpha$ real?

QII.A.3 Groups Group Homomorphisms and Isomorphisms View

What are the matrices directly orthogonally similar to $A$ if $A$ belongs to $\mathrm{SO}(2)$?

QII.A.4 Groups Group Homomorphisms and Isomorphisms View

What are the matrices directly orthogonally similar to $K_2 = \left(\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right)$?

QII.B.1 Groups Group Homomorphisms and Isomorphisms View

Show that $\left(\begin{array}{ll} 0 & 0 \\ 0 & 2 \end{array}\right)$ and $\left(\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right)$ are directly orthogonally similar.

QII.B.2 Groups Group Homomorphisms and Isomorphisms View

Show that $\left(\begin{array}{ll} 1 & 0 \\ 0 & 2 \end{array}\right)$ and $\left(\begin{array}{rr} 3 & 2 \\ -1 & 0 \end{array}\right)$ are similar but are not orthogonally similar.

QII.B.3 Groups Group Homomorphisms and Isomorphisms View

Show that $\left(\begin{array}{rr} 3 & 2 \\ -1 & 0 \end{array}\right)$ and its transpose are orthogonally similar but are not directly orthogonally similar.

QIII.A.1 Circles Circle Equation Derivation View

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$ and $(x,y)$ in $\mathbb{R}^2$, we denote by $\varphi_A(x,y)$ the determinant of the matrix $A(x,y) = \left(\begin{array}{cc} a-x & b-y \\ c+y & d-x \end{array}\right)$ and we consider $\mathcal{CP}_A$ the curve in $\mathbb{R}^2$ defined by the equation: $\varphi_A(x,y) = 0$.
Verify that $\mathcal{CP}_A$ is a circle (we agree that a circle can be reduced to a point); we will call $\mathcal{CP}_A$ the eigenvalue circle of $A$. Specify its center $C_A$ and its radius $r_A$.

QIII.A.2 Circles Circle-Line Intersection and Point Conditions View

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$, with $\varphi_A(x,y)$ the determinant of $A(x,y) = \left(\begin{array}{cc} a-x & b-y \\ c+y & d-x \end{array}\right)$ and $\mathcal{CP}_A$ the eigenvalue circle of $A$, specify, as a function of $A$, the cardinality of the intersection of $\mathcal{CP}_A$ with the $x$-axis $\mathbb{R} \times \{0\}$.

QIII.A.3 Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix View

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$, with $\varphi_A(x,y)$ the determinant of $A(x,y) = \left(\begin{array}{cc} a-x & b-y \\ c+y & d-x \end{array}\right)$, what do the solutions of the equation $\varphi_A(x,0) = 0$ represent for $A$? Specify the number of real eigenvalues of $A$ according to the value of $\Delta_A = (a-d)^2 + 4bc$.

QIII.B.1 Invariant lines and eigenvalues and vectors Compute or factor the characteristic polynomial View

Let $A \in \mathcal{M}_2(\mathbb{R})$. Compare the eigenvalue circle of $A$ and that of its transpose.

QIII.B.2 Invariant lines and eigenvalues and vectors Compute or factor the characteristic polynomial View

Let $A \in \mathcal{M}_2(\mathbb{R})$. Determine a necessary and sufficient condition on $\mathcal{CP}_A$ for $A$ to be symmetric.

QIII.B.3 Invariant lines and eigenvalues and vectors Geometric interpretation of eigenstructure View

Let $A \in \mathcal{M}_2(\mathbb{R})$.
a) Determine the matrices whose eigenvalue circle has zero radius and characterize geometrically their canonically associated endomorphism.
b) When the eigenvalue circle is reduced to its center, specify the canonically associated endomorphism, on the one hand when this center belongs to the unit circle (with center the origin $O=(0,0)$ and radius 1) and on the other hand when this center belongs to the $x$-axis.
c) What can be said about the matrix $A$ and $f_A$ when the eigenvalue circle $\mathcal{CP}_A$ has zero radius and center belonging to the $y$-axis $\{0\} \times \mathbb{R}$?

QIII.C Invariant lines and eigenvalues and vectors Geometric interpretation of eigenstructure View

Show that two matrices $A$ and $B$ of $\mathcal{M}_2(\mathbb{R})$ are directly orthogonally similar if and only if they have the same eigenvalue circle.

QIII.D.1 Invariant lines and eigenvalues and vectors Geometric interpretation of eigenstructure View

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$ we consider the four points (possibly coinciding) $E = (d,-c)$, $F = (a,b)$, $G = (d,b)$ and $H = (a,-c)$.
In the case where $A = \left(\begin{array}{rr} 1 & 7 \\ -1 & 3 \end{array}\right)$, draw the circle and the quadrilateral $EHFG$.

QIII.D.2 Invariant lines and eigenvalues and vectors Geometric interpretation of eigenstructure View

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$ we consider the four points (possibly coinciding) $E = (d,-c)$, $F = (a,b)$, $G = (d,b)$ and $H = (a,-c)$.
When the four points $E, F, G$ and $H$ are distinct show that they are the vertices of a rectangle, which we will call the eigenvalue rectangle of $A$.

QIII.D.3 Invariant lines and eigenvalues and vectors Geometric interpretation of eigenstructure View

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$ we consider the four points (possibly coinciding) $E = (d,-c)$, $F = (a,b)$, $G = (d,b)$ and $H = (a,-c)$.
Specify the matrices for which some of these points coincide, that is, when the rectangle is flattened.

QIII.E.1 Invariant lines and eigenvalues and vectors Geometric interpretation of eigenstructure View

Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$. Show that there exists a unique triplet $(\alpha, \beta, \gamma)$ of $\mathbb{R}^2 \times \mathbb{R}_+$ that we will specify, such that $A$ is directly orthogonally similar to $\left(\begin{array}{cc} \alpha+\gamma & -\beta \\ \beta & \alpha-\gamma \end{array}\right)$.

QIII.E.2 Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix View

Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$. According to the values of $(\alpha, \beta, \gamma)$ (where $A$ is directly orthogonally similar to $\left(\begin{array}{cc} \alpha+\gamma & -\beta \\ \beta & \alpha-\gamma \end{array}\right)$), specify the number of real eigenvalues of $A$.

QIII.E.3 Invariant lines and eigenvalues and vectors Geometric interpretation of eigenstructure View

Show that for every endomorphism $f$ of $\mathbb{R}^2$, there exist non-negative reals $k$ and $\ell$, a plane rotation $\rho_t$ and a reflection $\sigma_{t'}$ such that $f = k\rho_t + \ell\sigma_{t'}$.

QIII.E.4 Invariant lines and eigenvalues and vectors Geometric interpretation of eigenstructure View

Write a procedure or function in Maple or Mathematica that takes as input a quadruple $(a,b,c,d)$ of reals and returns a quadruple $(k, \ell, t, t')$ such that if $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ we have $f_A = k\rho_t + \ell\sigma_{t'}$.

QIV.A.1 Invariant lines and eigenvalues and vectors Diagonalizability determination or proof View

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, intersecting the $x$-axis. We denote by $L_1$ and $L_2$, with coordinates respectively $(\lambda_1, 0)$ and $(\lambda_2, 0)$, with $\lambda_1 < \lambda_2$, the two intersection points of $\mathcal{C}(\Omega, r)$ with the $x$-axis. Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$. We keep the notations $E, F, G, H$ from III.D.
Show that $A$ is diagonalizable.

QIV.A.2 Invariant lines and eigenvalues and vectors Compute eigenvectors or eigenspaces View

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, intersecting the $x$-axis. We denote by $L_1$ and $L_2$, with coordinates respectively $(\lambda_1, 0)$ and $(\lambda_2, 0)$, with $\lambda_1 < \lambda_2$, the two intersection points of $\mathcal{C}(\Omega, r)$ with the $x$-axis. Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$. We keep the notations $E, F, G, H$ from III.D.
Show that if $c \neq 0$, then $\left(\overrightarrow{L_1 E}, \overrightarrow{L_2 E}\right)$ is a basis of $\mathbb{R}^2$ consisting of eigenvectors for $f_A$.

QIV.A.3 Invariant lines and eigenvalues and vectors Compute eigenvectors or eigenspaces View

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, intersecting the $x$-axis. We denote by $L_1$ and $L_2$, with coordinates respectively $(\lambda_1, 0)$ and $(\lambda_2, 0)$, with $\lambda_1 < \lambda_2$, the two intersection points of $\mathcal{C}(\Omega, r)$ with the $x$-axis. Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$. We keep the notations $E, F, G, H$ from III.D.
When $c = 0$, can we give a basis of eigenvectors for $f_A$ using the eigenvalue circle and the eigenvalue rectangle?

QIV.A.4 Invariant lines and eigenvalues and vectors Compute eigenvectors or eigenspaces View

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, intersecting the $x$-axis. We denote by $L_1$ and $L_2$, with coordinates respectively $(\lambda_1, 0)$ and $(\lambda_2, 0)$, with $\lambda_1 < \lambda_2$, the two intersection points of $\mathcal{C}(\Omega, r)$ with the $x$-axis. Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$. We keep the notations $E, F, G, H$ from III.D.
Show that the square of the cosine of the angle between two eigenvectors of $A$ associated with two distinct eigenvalues is determined by the circle $\mathcal{C}(\Omega, r)$, and does not depend on the choice of a matrix $A$ whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$ (one may, if deemed useful, introduce the orthogonal projection of $\Omega$ onto the $x$-axis). What about if $A$ is symmetric?

QIV.A.5 Invariant lines and eigenvalues and vectors Geometric interpretation of eigenstructure View

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, intersecting the $x$-axis. Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$.
Characterize geometrically $f_A$ when $\Omega = O$, with $O = (0,0)$, and $r = 1$.

QIV.A.6 Invariant lines and eigenvalues and vectors Geometric interpretation of eigenstructure View

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, intersecting the $x$-axis. Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$.
Characterize geometrically $f_A$ when $\mathcal{CP}_A$ is the circle with diameter the segment $[O, I]$ with $I = (1,0)$.

QIV.B.1 Invariant lines and eigenvalues and vectors Diagonalizability determination or proof View

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, tangent to the $x$-axis. We call $L$, with coordinates $(\lambda, 0)$, the point of tangency of $\mathcal{C}(\Omega, r)$ with the $x$-axis. Let $A$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$.
Is the matrix $A$ diagonalizable? Is it trigonalizable?

QIV.B.2 Invariant lines and eigenvalues and vectors Compute eigenvectors or eigenspaces View

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, tangent to the $x$-axis. We call $L$, with coordinates $(\lambda, 0)$, the point of tangency of $\mathcal{C}(\Omega, r)$ with the $x$-axis. Let $A$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$. We keep the notations $E, F, G, H$ from III.D.
Can we give an eigenvector using the points $L, E, F, G$ and $H$?

QIV.B.3 Invariant lines and eigenvalues and vectors Spectral properties of structured or special matrices View

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, tangent to the $x$-axis. Let $A$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$.
What can be said about matrices whose eigenvalue circle is tangent to the $x$-axis and whose center is located on the $y$-axis?

QIV.B.4 Invariant lines and eigenvalues and vectors Diagonalize a matrix explicitly View

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, tangent to the $x$-axis. We call $L$, with coordinates $(\lambda, 0)$, the point of tangency of $\mathcal{C}(\Omega, r)$ with the $x$-axis. Let $A$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$.
Show that there exists a unique non-zero real $\alpha$ such that $A$ is directly orthogonally similar to the matrix $T_{\lambda,\alpha} = \left(\begin{array}{cc} \lambda & \alpha \\ 0 & \lambda \end{array}\right)$. Specify $\alpha$ using the elements of the matrix $A$. Where can we find this number on the eigenvalue circle?

QIV.B.5 Invariant lines and eigenvalues and vectors Compute eigenvectors or eigenspaces View

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, tangent to the $x$-axis. We call $L$, with coordinates $(\lambda, 0)$, the point of tangency of $\mathcal{C}(\Omega, r)$ with the $x$-axis. Let $A$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$, and let $\alpha$ be the unique non-zero real such that $A$ is directly orthogonally similar to $T_{\lambda,\alpha} = \left(\begin{array}{cc} \lambda & \alpha \\ 0 & \lambda \end{array}\right)$.
Show that there exists an orthonormal direct basis $(e_1, e_2)$ of the plane such that for all $u$ in $\mathbb{R}^2$, we have $f_A(u) = \lambda u + \alpha (e_2 \mid u) e_1$.

QIV.C.1 Invariant lines and eigenvalues and vectors Diagonalizability determination or proof View

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and radius $r \geqslant 0$ disjoint from the $x$-axis. We denote by $K$ the orthogonal projection of $\Omega$ onto the $x$-axis. Let $A$ be a matrix with proper eigenvalue circle equal to $\mathcal{C}(\Omega, r)$.
Does there exist a matrix $P$ in $\mathrm{GL}_2(\mathbb{R})$ such that the matrix $P^{-1}AP$ is diagonal? Does there exist a matrix $P$ in $\mathrm{GL}_2(\mathbb{R})$ such that the matrix $P^{-1}AP$ is upper triangular?

QIV.C.2 Invariant lines and eigenvalues and vectors Geometric interpretation of eigenstructure View

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and radius $r \geqslant 0$ disjoint from the $x$-axis. We denote by $K$ the orthogonal projection of $\Omega$ onto the $x$-axis. Let $A$ be a matrix with proper eigenvalue circle equal to $\mathcal{C}(\Omega, r)$.
Determine the points of $\mathcal{C}(\Omega, r)$ at which the tangent line to $\mathcal{C}(\Omega, r)$ contains $K$.

QIV.C.3 Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix View

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and radius $r \geqslant 0$ disjoint from the $x$-axis. We denote by $K$ the orthogonal projection of $\Omega$ onto the $x$-axis. Let $A$ be a matrix with proper eigenvalue circle equal to $\mathcal{C}(\Omega, r)$. Let $U$ be one of the points of $\mathcal{C}(\Omega, r)$ at which the tangent line contains $K$.
Express the eigenvalues of $A$, considered as an element of $\mathcal{M}_2(\mathbb{C})$, using the abscissa of $K$ and the distance $KU$ from $K$ to $U$.

QIV.D.1 Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix View

In this section, we consider in $\mathbb{R}^2$ a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and radius $r$ and $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ a matrix with proper eigenvalue circle equal to $\mathcal{C}(\Omega, r)$.
In this question, $\Omega = (\alpha, \beta) \in \mathbb{R} \times \mathbb{R}^*$, $r = |\beta|$ and $E = (\alpha + |\beta|, \beta)$.
Specify the eigenvalues of $A$ and give a matrix $B$ whose off-diagonal entries are opposite and which is directly orthogonally similar to $A$, as well as an orthogonal decomposition of the endomorphism canonically associated with $B$.

QIV.D.2 Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix View

In this section, we consider in $\mathbb{R}^2$ a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and radius $r$ and $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ a matrix with proper eigenvalue circle equal to $\mathcal{C}(\Omega, r)$.
In this question $\Omega = (0, \alpha)$ with $\alpha > 0$ and $r = \alpha/2$.
Specify the eigenvalues of $A$ and give a matrix $B$ whose off-diagonal entries are opposite and which is directly orthogonally similar to $A$, as well as an orthogonal decomposition of the endomorphism canonically associated with $B$. Make a drawing in the case where $\alpha = 6$ illustrating questions IV.C.2 and IV.C.3.

QV.A.1 Invariant lines and eigenvalues and vectors Compute or factor the characteristic polynomial View

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$ and $(x,y,z)$ in $\mathbb{R}^3$, we denote by $\psi_A(x,y,z)$ the real part of the determinant of the matrix $\left(\begin{array}{cc} a-x-\mathrm{i}z & b-y \\ c+y & d-x-\mathrm{i}z \end{array}\right)$, where $\mathrm{i}$ is the complex affix of the point $J = (0,1)$.
Calculate $\psi_A(x,y,z)$.

QV.A.2 Invariant lines and eigenvalues and vectors Geometric interpretation of eigenstructure View

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$ and $(x,y,z)$ in $\mathbb{R}^3$, we denote by $\psi_A(x,y,z)$ the real part of the determinant of the matrix $\left(\begin{array}{cc} a-x-\mathrm{i}z & b-y \\ c+y & d-x-\mathrm{i}z \end{array}\right)$, where $\mathrm{i}$ is the complex affix of the point $J = (0,1)$.
Specify the nature of the quadric $\mathcal{H}_A$ with equation $\psi_A(x,y,z) = 0$.

QV.B.1 Invariant lines and eigenvalues and vectors Geometric interpretation of eigenstructure View

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$, let $\mathcal{H}_A$ be the quadric with equation $\psi_A(x,y,z) = 0$ where $\psi_A(x,y,z)$ is the real part of the determinant of $\left(\begin{array}{cc} a-x-\mathrm{i}z & b-y \\ c+y & d-x-\mathrm{i}z \end{array}\right)$.
Specify the intersection of $\mathcal{H}_A$ with the plane with equation $z = 0$.

QV.B.2 Conic sections Locus and Trajectory Derivation View

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$, let $\mathcal{H}_A$ be the quadric with equation $\psi_A(x,y,z) = 0$ where $\psi_A(x,y,z)$ is the real part of the determinant of $\left(\begin{array}{cc} a-x-\mathrm{i}z & b-y \\ c+y & d-x-\mathrm{i}z \end{array}\right)$.
Specify the intersection $Z_A$ of $\mathcal{H}_A$ with the plane with equation $x = (a+d)/2$.

QV.C.1 Conic sections Locus and Trajectory Derivation View

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$, let $\mathcal{H}_A$ be the quadric with equation $\psi_A(x,y,z) = 0$ where $\psi_A(x,y,z)$ is the real part of the determinant of $\left(\begin{array}{cc} a-x-\mathrm{i}z & b-y \\ c+y & d-x-\mathrm{i}z \end{array}\right)$, and let $Z_A$ be the intersection of $\mathcal{H}_A$ with the plane $x = (a+d)/2$.
If the matrix $A$ has two non-real eigenvalues, how can one see the eigenvalues of $A$ on $\mathcal{H}_A$? (One may consider the intersection of $Z_A$ with the plane with equation $y = 0$.) Can one see a basis of eigenvectors using $\mathcal{H}_A$?

QV.C.2 Conic sections Locus and Trajectory Derivation View

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$, let $\mathcal{H}_A$ be the quadric with equation $\psi_A(x,y,z) = 0$ where $\psi_A(x,y,z)$ is the real part of the determinant of $\left(\begin{array}{cc} a-x-\mathrm{i}z & b-y \\ c+y & d-x-\mathrm{i}z \end{array}\right)$.
In the case where $A = \left(\begin{array}{rr} 1 & 7 \\ -1 & 3 \end{array}\right)$ make a perspective drawing illustrating what precedes.

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