Digest for "Diagonalisation in the Field of Complex Numbers"

Assume that \( (x_{1},y_{1},x_{2},y_{2})\in\mathbb{R}^{4}\) are real numbers, and consider the complex numbers \( z_1=x_1+iy_1\) and \( z_2=x_2+iy_2\) .

Then we define the sum and the difference of \( z_{1}\) and \( z_{2}\) as:

• \( z_{1}+z_{2}=(x_{1}+x_{2})+i(y_{1}+y_{2})\) ,

• and \( z_{1}-z_{2}=(x_{1}-x_{2})+i(y_{1}-y_{2})\) .

Assume that \( (z_{1},z_{2}\in\mathbb{C}^{2}\) are complex numbers with \( z_2=x_2+iy_2\) , wher \( (x_{2},y_{2})\in\mathbb{R}^{2}\) are real numbers.

Consider the opposite of of \( z_{2}\) , \( -z_2=(-x_2)+i(-y_2)\) .

Then \( z_{1}-z_{2}=z_{1}+(-z_{2})\) .

Assume that \( (z_1,z_2,z_3)\in\mathbb{C}^3\) are complex numbers.

Then the following properties hold for the addition.

1. Commutativity: \( z_1+z_2=z_2+z_1\) .

2. Associativity: \( (z_1+z_2)+z_3=z_1+(z_2+z_3)\) .

3. \( 0=0+0i\) is neutral: \( z_1+0=0+z_1=z_1\)

4. \( z_1+(-z_1)=(-z_1)+z_1=0\) , which justifies the fact that we call \( -z_{1}\) the opposite of \( z_{1}\) .

Assume that \( (x_{1},y_{1},x_{2},y_{2})\in\mathbb{R}^{4}\) are real numbers, and consider the complex numbers \( z_1=x_1+iy_1\) and \( z_2=x_2+iy_2\) .

Then we define the product and the quotient (if \( z_{2}\ne 0\) ) of \( z_{1}\) and \( z_{2}\) as:

• \( z_1z_2=(x_1x_2-y_1y_2)+i(x_1y_2+y_1x_2)\) ,

• and, if \( z_{2}\ne 0\) , \( \frac{z_1}{z_2}=\frac{x_1x_2+y_1y_2}{x_2^2+y_2^2} +i\;\frac{-x_1y_2+x_2y_1}{x_2^2+y_2^2}\) .

Assume that \( (z_{1},z_{2})\in\mathbb{C}\times\mathbb{C}^{*}\) are complex numbers such that \( z_{2}\ne 0\) . with \( z_2=x_2+iy_2\) , where \( (x_{2},y_{2})\in\mathbb{R}^{2}\) are real numbers.

Consider the inverse of of \( \frac{1}{z_2}=\frac{x_2}{x_2^2+y_2^2}+i\;\frac{-y_2}{x_2^2+y_2^2}\) .

Then \( \frac{z_1}{z_2}=z_1\frac{1}{z_2}\) .

Assume that \( (z_1,z_2)\in(\mathbb{C}^*)^2\) are non-zero complex numbers, and assume that \( z_1=R_1e^{i\theta_1}\) and \( z_2=R_2e^{i\theta_2}\) , with \( (R_{1},R_{2})\in(\mathbb{R}^{*})^{2}\) the modules of \( z_{1}\) and \( z_{2}\) respectively, and \( (\theta_1,\theta_2)\in\mathbb{R}^{2}\) are the arguments of \( z_{1}\) and \( z_{2}\) respectively.

Then \( z_1z_2=R_1R_2e^{i(\theta_1+\theta_2)}\) and \( \frac{z_{1}}{z_{2}}=\frac{R_{1}}{R_{2}}e^{i(\theta_{1}-\theta_{2})}\) .

Assume that \( (z_1,z_2,z_3)\in(\mathbb{C})^{*})^3\) are non zero complex numbers.

Then the following properties hold for the multiplication.

1. Commutativity: \( z_1z_2=z_2z_1\) .

2. Associativity: \( (z_1z_2)z_3=z_1(z_2z_3)\) .

3. \( 1=1e^{i0}i\) is neutral: \( 1\times z_1=z_1\times 1=z_1\)

4. \( z_1\frac{1}{z_1}=\frac{1}{z_1}\times z_1=1\) , which justifies the fact that we call \( \frac{1}{z_1}\) the inverse of \( z_{1}\) .

Assume that \( (z_1,z_2,z_3)\in\mathbb{C}^3\) are complex numbers.

Then the following properties hold for the multiplication.

1. Commutativity: \( z_1z_2=z_2z_1\) .

2. Associativity: \( (z_1z_2)z_3=z_1(z_2z_3)\) .

3. \( 1=1+0i\) is neutral: \( 1\times z_1=z_1\times 1=z_1\) .

Assume that \( z\in\mathbb{C}\) is a complex numbers.

Then \( 0=0+0i\) is absorbent for the multiplication, that is:

• \( 0\times z=z\times 0=0\) .

Assume that \( (z_0,z_1,z_2)\in\mathbb{C}^3\) are any complex numbers.

Then the following properties hold for the addition and multiplication.

1. Distributivity to the left of the multiplication on the addition: \( z_0(z_1+z_2)=z_0z_1+z_0z_2\) .

2. Distributivity to the right of the multiplication on the addition: \( (z_1+z_2)z_0=z_1z_0+z_2z_0\)

Assume that \( z\in\mathbb{C}\) is a complex number, with \( z=x+iy\) , where \( (x,y\in\mathbb{R}^{2}\) are real numbers.

Then the complex conjugate of is defined as \( \overline{z}=x-iy\)

Assume that \( (z_1,z_2)\in\mathbb{C}^2\) are complex numbers, and that \( x\in\mathbb{R}\) is a real number, considered as the complex number \( x+0i\) .

Then the following equalities hold.

1. \( \overline{x}=x\) .

2. \( \overline{xz_1}=x\overline{z_1}\) .

3. \( \overline{\overline{z_1}}=z_1\) , so that we may speak of a pair of conjugate complex numbers.

4. \( \overline{z_1+z_2}=\overline{z_1}+\overline{z_2}\) .

5. \( \overline{-z_1}=-\overline{z_1}\) .

6. \( \overline{z_1-z_2}=\overline{z_1}-\overline{z_2}\) .

7. \( \overline{z_1z_2}=\overline{z_1}\;\overline{z_2}\)

8. If \( z_1\ne 0\) , then \( \overline{z_1}\ne 0\) and \( \overline{\left(\frac{1}{z_1}\right)}=\frac{1}{\overline{z_1}}\) .

9. If \( z_2\ne 0\) , then \( \overline{z_2}\ne 0\) and \( \overline{\left(\frac{z_{1}}{z_2}\right)}=\frac{\overline{z_1}}{\overline{z_2}}\) .

10. \( z_{1}\overline{z_{1}}\) is a positive or zero real number, and the module of \( z_{1}\) is equal to \( |z_{1}|=\sqrt{z_{1}\overline{z_{1}}}\) .

11. If \( z_1\ne 0\) and \( z_{1}=Re^{i\theta}\) , with \( R=|z_{1}|\) and \( \theta=\text{arg}(z_{1})\) , then \( \overline{z_1}=Re^{-i\theta}\) .

Assume that \( a\in\mathbb{R}\) is a real number of any sign.

Then there exists at least one complex square root of \( a\) , a complex number \( z\in\mathbb{C}\) such that \( z^{2}=a\) .

Namely, these are the following complex numbers.

• If \( a=0\) , it has a unique complex square root \( z_{0}=0\) .

• If \( a>0\) , it has two complex square roots, that are the real numbers \( z_1=\sqrt{a}\) and \( z_2=-\sqrt{a}\) .

• If \( a<0\) , it has two complex square roots, that are the conjugate complex numbers \( z_1=i\sqrt{-a}\) and \( z_2=-i\sqrt{-a}\) .

Assume that \( (a,b,c,d)\in\mathbb{C}^4\) are complex numbers.

Then the \( 2\times 2\) matrix with complex elements \( (a,b,c,d)\) is:

• \( A=\begin{bmatrix}a&b\\ c&d\end{bmatrix}\)

If \( (a,b,c,d)\in\mathbb{R}^4\) are real numbers, is is also a \( 2\times 2\) matrix with real elements.

Assume that \( (a,b,c,d,e,f,g,h)\in\mathbb{C}^8\) are complex numbers.

Consider the matrices with complex elements \( A=\begin{bmatrix}a&b\\ c&d\end{bmatrix}\) and \( B=\begin{bmatrix}e&f\\ g&h\end{bmatrix}\) .

Then the sum \( S=A+B\) of \( A\) and \( B\) is defined as the \( 2\times 2\) matrix with complex elements:

• \( S=\begin{bmatrix}a+e&b+f\\ c+g&d+h\end{bmatrix}\)

And the difference \( S=A-B\) of \( A\) and \( B\) is defined as the \( 2\times 2\) matrix with complex elements:

• \( D=\begin{bmatrix}a-e&b-f\\ c-g&d-h\end{bmatrix}\)

Assume that \( (A,B,C)\in(\mathbb{M}_{22}^{\mathbb{C}})^3\) are \( 2\times 2\) matrices with complex elements, with \( A=\begin{bmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{bmatrix}\) .

Then \( (\mathbb{M}_{22}^{\mathbb{C}}\;,+)\) is a commutative group because of the following properties of the addition of \( 2\times 2\) matrices with complex elements.

1. Commutativity: \( A+B=B+A\) .

2. Associativity: \( (A+B)+C=A+(B+C)\) .

3. The null matrix \( O_{22}=\begin{bmatrix}0&0\\ 0&0\end{bmatrix}\) is neutral: \( A+O_{22}=O_{22}+A=A\) .

4. If we define the opposite \( -A\) of \( A\) as \( -A=\begin{bmatrix}-a_{11}&-a_{12}\\ -a_{21}&-a_{22}\end{bmatrix}\) , then \( A+(-A)=(-A)+A=O_{22}\) .

Assume that \( A\in\mathbb{M}_{22}^{\mathbb{C}}\) is a \( 2\times 2\) matrix with complex elements.

Then the following assertions hold for the subtraction.

1. \( A-O_{22}=A\) .

2. \( O_{22}-A=-A\) .

3. \( A-A=O_{22}\) .

Assume that \( z\in\mathbb{C}\) is a complex number.

Then the following assertions hold for the subtraction.

1. \( z-0=z\) .

2. \( 0-z=-z\) .

3. \( z-z=0\) .

Assume that \( (A,B)\in(\mathbb{M}_{22}^{\mathbb{C}})^2\) are \( 2\times 2\) matrices with complex elements. Then the following assertions hold for the addition and the subtraction.

1. Subtract a matrix is adding its opposite: \( A-B=A+(-B)\) .

2. The addition and subtraction of matrices are mutually reciprocal:

1. \( (A+B)-B=A\) .

2. \( (A-B)+B=A\) .

Assume that \( (z_{1},z_{2})\in(\mathbb{C})^2\) are complex numbers.

Then the following assertions hold for the addition and the subtraction.

1. Subtract a complex number is adding its opposite: \( z_{1}-z_{2}=z_{1}+(-z_{2})\) .

2. The addition and subtraction of complex numbers are mutually reciprocal:

1. \( (z_{1}+z_{2})-z_{2}=z_{1}\) .

2. \( (z_{1}-z_{2})+z_{2}=z_{1}\) .

Assume that \( (a,b,c,d,k)\in\mathbb{C}^5\) are complex numbers.

Consider the \( 2\times 2\) matrix with complex elements \( A=\begin{bmatrix}a&b\\ c&d\end{bmatrix}\) .

Then the product \( P=kA\) of \( A\) by the complex scalar \( k\) is defined as the \( 2\times 2\) matrix with complex elements:

• \( P=\begin{bmatrix}ka&kb\\ kc&kd\end{bmatrix}\)

Assume that \( A\in\mathbb{M}_{22}^{\mathbb{C}}\) is a \( 2\times 2\) matrix with complex elements and that \( k\in\mathbb{C}\) is a complex scalar.

Then the following properties hold for the scalar multiplication.

1. \( 1\times A=A\) .

2. \( 0\times A=O_{22}\) .

3. \( k\times O_{22}=O_{22}\)

4. \( k(-A)=-kA\)

5. \( (-k)A=-kA\)

Assume that \( (z_{1},z_{2})\in(\mathbb{C})^2\) are complex numbers.

Then the following assertions hold for the addition and the subtraction.

1. \( z_{1}(-z_{2})=-z_{1}z_{2}\) .

2. \( (-z_{1})z_{2}=-z_{1}z_{2}\) .

Assume that \( (A,B)\in(\mathbb{M}_{22}^{\mathbb{C}})^2\) are \( 2\times 2\) matrices with complex elements and that \( (k,l)\in\mathbb{C}^2\) are complex scalars.

Then \( (\mathbb{M}_{22}^{\mathbb{C}}\;,+,\cdot)\) is a vector space on \( \mathbb{C}\) because of the following properties of the addition and the scalar multiplication.

1. \( (\mathbb{M}_{22}^{\mathbb{C}}\;,+)\) is a commutative group.

2. First distributivity law: \( k(A+B)=kA+kB\) .

3. Second distributivity law: \( (k+l)A=kA+lA\) .

4. Associativity law: \( k(lA)=(kl)A\) .

Assume that \( (a,b,c,d,e,f,g,h)\in\mathbb{C}^8\) are complex numbers.

Consider the matrices with complex elements \( A=\begin{bmatrix}a&b\\ c&d\end{bmatrix}\) and \( B=\begin{bmatrix}e&f\\ g&h\end{bmatrix}\) .

Then the sum result of the matrix multiplication \( M=AB\) of \( A\) and \( B\) is defined as the \( 2\times 2\) matrix with complex elements:

• \( M=\begin{bmatrix}ae+bg&af+bh\\ ce+dg&cf+dh\end{bmatrix}\)

Assume that \( (A,B,C)\in(\mathbb{M}_{22}^{\mathbb{C}})^3\) are \( 2\times 2\) matrices with complex elements.

Then \( (\mathbb{M}_{22}^{\mathbb{C}}\;,\times)\) is a unitary monoid because of the following properties of the addition of \( 2\times 2\) matrices with complex elements.

1. Associativity: \( (AB)C=A(BC)\) .

2. The identity matrix \( I=\begin{bmatrix}1&0\\ 0&1\end{bmatrix}\) is neutral: \( IA=AI=A\) .

Moreover, the following property holds.

• The null matrix \( O_{22}=\begin{bmatrix}0&0\\ 0&0\end{bmatrix}\) is absorbent: \( AO_{22}=O_{22}A=O_{22}\) .

Assume that \( (A,B,C)\in(\mathbb{M}_{22}^{\mathbb{C}})^3\) are \( 2\times 2\) matrices with complex elements.

Then \( (\mathbb{M}_{22}^{\mathbb{C}}\;,+,\times)\) is a unitary ring because of the following properties of the matrix multiplication and the addition of \( 2\times 2\) matrices with complex elements.

1. \( (\mathbb{M}_{22}^{\mathbb{C}}\;,+)\) is a commutative group.

2. \( (\mathbb{M}_{22}^{\mathbb{C}}\;,\times)\) is a unitary monoid.

3. Distributivity to the left: \( A(B+C)=AB+AC\) .

4. Distributivity to the right: \( (A+B)C=AC+BC\) .

Assume that \( (A,B)\in(\mathbb{M}_{22}^{\mathbb{C}})\) are \( 2\times 2\) matrices with complex elements and that \( (k,l)\in\mathbb{C}^2\) are complex scalars.

Then \( (\mathbb{M}_{22}^{\mathbb{C}}\;,+,\times,\cdot)\) is a unitary algebra because of the following properties of the matrix multiplication, the addition and the scalar multiplication of \( 2\times 2\) matrices with complex elements.

1. \( (\mathbb{M}_{22}^{\mathbb{C}}\;,+,\cdot)\) is a vector space on \( \mathbb{C}\) .

2. \( (\mathbb{M}_{22}^{\mathbb{C}}\;,+,\times)\) is a unitary ring.

3. Associativity law: \( (kA)(lB)=(kl)(AB)\) .

Assume that \( (a,b,c,d)\in\mathbb{C}^4\) are complex numbers.

Consider the \( 2\times 2\) matrix with complex elements \( A=\begin{bmatrix}a&b\\ c&d\end{bmatrix}\) .

Then the determinant of the matrix \( A\) is defined as the complex number \( \det(A)=ad-bc\) .

And if \( \det(A)\ne 0\) , then \( A\) is said to be invertible and its inverse is defined as \( A^{-1}=\frac{1}{ad-cb}\begin{bmatrix}d&-b\\ -c&a\end{bmatrix}\) .

Assume that \( (A,B)\in(\mathbb{M}_{22}^{\mathbb{C}})\) are \( 2\times 2\) matrices with complex elements such that \( \det(A)\ne 0\) and \( \det(B)\ne 0\) .

Then the following assertions hold for the inverses of matrices.

1. \( A^{-1}A=AA^{-1}=I\) , the identity matrix.

2. \( \det(AB)\ne 0\) and \( (AB)^{-1}=B^{-1}A^{-1}\) .

Assume that \( (A,B)\in(\mathbb{M}_{22}^{\mathbb{C}})\) are \( 2\times 2\) matrices with complex elements. Then \( \det(AB)=\det(A)\det(B)\) .

Assume that \( (e,f)\in\mathbb{C}^2\) are complex numbers.

Then the column vector with \( 2\) complex elements \( (e,f)\) is:

• \( U=\begin{bmatrix}e\\ f\end{bmatrix}\) .

If \( (e,f)\in\mathbb{R}^2\) are real numbers, it is also a column vector with \( 2\) real elements.

Assume that \( (e,f,g,h)\in\mathbb{C}^4\) are complex numbers.

Then the sum \( W_{+}=U+V\) of \( U\) and \( V\) is defined as the column vector with \( 2\) complex elements:

• \( W_+=\begin{bmatrix}e+g\\ f+h\end{bmatrix}\)

And the difference \( W_{-}=U-V\) of \( U\) and \( V\) is defined as the column vector with \( 2\) complex elements:

• \( W_-=\begin{bmatrix}e-g\\ f-h\end{bmatrix}\)

Assume that \( (U,V,W)\in(\mathbb{M}_{21}^{\mathbb{C}})^3\) are column vectors with \( 2\) complex elements, with \( U=\begin{bmatrix}e\\ f\end{bmatrix}\) .

Then \( (\mathbb{M}_{21}^{\mathbb{C}}\;,+)\) is a commutative group because of the following properties of the addition of column vectors with \( 2\) complex elements.

1. Commutativity: \( U+V=V+U\) .

2. Associativity: \( (U+V)+W=U+(V+W)\)

3. The null column vector \( O_{21}=\begin{bmatrix}0\\ 0\end{bmatrix}\) is neutral: \( U+O_{21}=O_{21}+U=U\) .

4. If we define the opposite \( -U\) of \( U\) as \( -U=\begin{bmatrix}-e\\ -f\end{bmatrix}\) , then \( U+(-U)=(-U)+U=O_{21}\) .

Assume that \( (e,f,k)\in\mathbb{C}^3\) are complex numbers.

Consider the column vector with \( 2\) complex elements \( U=\begin{bmatrix}e\\ f\end{bmatrix}\) .

Then the product \( T=kU\) of \( U\) by the complex scalar \( k\) is defined as the column vector with \( 2\) complex elements:

• \( T=\begin{bmatrix}ke\\ kf\end{bmatrix}\) .

Assume that \( U\in\mathbb{M}_{21}^{\mathbb{C}}\) is a column vector with \( 2\) complex elements and that \( k\in\mathbb{C}\) is a complex scalar.

Then the following properties hold for the scalar multiplication.

1. \( 1\times U=U\) .

2. \( 0\times U=O_{21}\) .

3. \( k\times O_{21}=O_{21}\)

4. \( k(-A)=-kA\)

5. \( (-k)A=-kA\)

Assume that \( (U,V)\in(\mathbb{M}_{21}^{\mathbb{C}})^2\) are column vectors with \( 2\) complex elements and that \( (k,l)\in\mathbb{C}^2\) are complex scalars.

Then \( (\mathbb{M}_{21}^{\mathbb{C}}\;,+,\cdot)\) is a vector space on \( \mathbb{C}\) because of the following properties of the addition and the scalar multiplication.

1. \( (\mathbb{M}_{21}^{\mathbb{C}}\;,+)\) is a commutative group.

2. First distributivity law: \( k(U+V)=kU+kV\) .

3. Second distributivity law: \( (k+l)U=kU+lU\) .

4. Associativity law: \( k(lU)=(kl)U\) .

Assume that \( (a,b,c,d,e,f)\in\mathbb{C}^8\) are complex numbers.

Consider the \( 2\times 2\) matrix with complex elements \( A=\begin{bmatrix}a&b\\ c&d\end{bmatrix}\) and the column vector with \( 2\) complex elements \( U=\begin{bmatrix}e\\ f\end{bmatrix}\) .

Then the result of the matrix multiplication \( H=AU\) is defined as the column vector with complex elements:

• \( H=\begin{bmatrix}ae+bf\\ ce+df\end{bmatrix}\) .

Assume that \( (A,B)\in(\mathbb{M}_{22}^{\mathbb{C}})^2\) are \( 2\times 2\) matrices with complex elements and that \( U\in\mathbb{M}_{21}^{\mathbb{C}}\) is a column vector with \( 2\) complex elements.

Then the following properties hold.

1. Associativity: \( (AB)U=A(BU)\) .

2. The identity matrix \( I=\begin{bmatrix}1&0\\ 0&1\end{bmatrix}\) is neutral: \( IU=U\) .

3. The null matrix \( O_{22}=\begin{bmatrix}0&0\\ 0&0\end{bmatrix}\) is absorbent: \( O_{22}U=O_{21}\) .

4. The null column vector \( O_{21}=\begin{bmatrix}0\\ 0\end{bmatrix}\) is absorbent: \( AO_{21}=O_{21}\) .

Assume that \( (a,b,c,d,e,f)\in\mathbb{C}^6\) are complex numbers.

Then solving the system of \( 2\) linear equations of \( 2\) complex variables:

is finding the couple(s) of complex numbers \( (x,y)\in\mathbb{C}^2\) that fulfil both equations of the system together.

Assume \( (a,b,c,d,e,f)\in\mathbb{C}^6\) are real numbers such that \( a\) and \( b\) are not zero together, and \( c\) and \( d\) are not zero together. Then the system of 2 linear equations of 2 real variables:

has a unique solution in \( (x,y)\in\mathbb{R}^2\) if and only if \( ad-bc\neq 0\) , and that solution is:

Assume \( (a,b,c,d,e,f)\in\mathbb{C}^6\) are real numbers such that

\( ad-bc=0\) . Then the set of solutions of the system of 2 linear equations of 2 real variables:

is either an infinite set, or the void set \( \emptyset\) .

Assume that \( (a,b,c,d,e,f)\in\mathbb{C}^6\) are complex numbers, and consider:

• the \( 2\times 2\) matrix with complex elements \( A=\begin{bmatrix}a&b\\ c&d\end{bmatrix}\) ,

• and the column vector with \( 2\) complex elements \( B=\begin{bmatrix}e\\ f\end{bmatrix}\) .

Then the \( 2\times 2\) linear system in the complex variables \( (x,y)\in\mathbb{C}^{2}\) :

is equivalent to the matrix equation \( AX=B\) in the unknown column

vector \( X=\begin{bmatrix}x\\ y\end{bmatrix}\) .

Assume that \( A\in\mathbb{M}_{22}^{\mathbb{C}}\) is a \( 2\times 2\) matrix with complex elements such that \( \det(A)\ne 0\) , and that \( B\in\mathbb{M}_{21}^{\mathbb{C}}\) is a column vector with \( 2\) complex elements.

Then the matrix equation in the column vector with \( 2\) complex elements \( X\in\mathbb{M}_{21}^{\mathbb{C}}\) :

has a unique solution:

Assume that \( (a,b,c,d)\in\mathbb{C}^4\) are complex numbers, and consider the matrix \( A=\begin{bmatrix}a&b\\ c&d\end{bmatrix}\) and the null column vector \( O_{21}=\begin{bmatrix}0\\ 0\end{bmatrix}\) .

Then the solution(s) of the matrix equation \( AX=O_{21}\) is (are) the following.

• If \( \det(A)\ne 0\) , then the matrix equation has a unique solution, and it is \( X=O_{21}=\begin{bmatrix}0\\ 0\end{bmatrix}\) .

• If \( A\ne O_{22}\) and \( \det(A) = 0\) , then the solutions are the column vectors that are the scalar multiples of \( \begin{bmatrix}-b\\ a\end{bmatrix}\) , unless \( a=b=0\) , in which case the solutions are the column vectors that are the scalar multiples of \( \begin{bmatrix}-d\\ c\end{bmatrix}\) .

• And if \( A=O_{22}\) the solutions are all the column vectors \( X=\begin{bmatrix}x\\ y\end{bmatrix}\) , for any \( (x,y)\in\mathbb{C}^2\) .

Assume that \( (a,b,c,d)\in\mathbb{R}^4\) are real numbers and consider the homogeneous system:

Then the solution(s) of the homogeneous system is (are) the following.

1. If \( ad-bc\ne 0\) , then the system has a unique solution, and it is

   \( (x,y)=(0,0)\) .

2. If \( a\) , \( b\) , \( c\) and \( d\) are not all zeros and \( ad-bc=0\) , then the solutions are the couples of complex numbers that are proprtionnal to \( (-b,a)\) , unless \( a=b=0\) , in which case the solutions are couples of complex numbers that are \( (-d,c)\) .

3. And if \( a=b=c=d=0\) the solutions are all the elements of \( \mathbb{R}^2\) .