Linear algebra

Definition. Let M={1,2,…,m} and N={1,2,…,n}. A map , $A(i,j)={{\left( {{a}_{ij}} \right)}_{\begin{smallmatrix} i=\overline{1,m} \\ j=\overline{1,n} \end{smallmatrix}}}$ is called a matrix of type $(m,n);$ with m rows and n columns:

$A=\left( \begin{aligned} & {{a}_{11}}\text{ }{{a}_{12}} ... {{a}_{1n}} \\ & {{a}_{21}}\text{ }{{a}_{22}} ... {{a}_{2n}} \\ & ...\text{ }...\text{ }...\text{ }... \\ & {{a}_{m1}}\text{ }{{a}_{m2}} ... {{a}_{mn}} \\ \end{aligned} \right)$ and we write for the set of matrices of type $(m,n)$ with complex entries.

If then the matrix is called square of order $n$, and the set of such matrices is written ${{M}_{n}}(\mathbb{C})$.

Operations with matrices:

1. Addition

Let , then $C=A+B\in {{M}_{m,n}}(\mathbb{C})$ where is their sum.

Properties, $(\forall )A,B,C\in {{M}_{m,n}}(\mathbb{C})$:

• (commutativity);

• (associativity);

• $A+O=O+A=A$ (the identity element is the zero matrix $O$);

• $A+(-A)=(-A)+A=O$ (the opposite of is ).

2. Multiplication by scalars

Let and , then $B=\lambda \cdot A\in {{M}_{m,n}}(\mathbb{C})$, where ${{b}_{ij}}=\lambda \cdot {{a}_{ij}}$ is the product of the matrix A with the scalar .

Properties, $(\forall )A,B\in {{M}_{m,n}}(\mathbb{C})$ and $\lambda ,\mu \in \mathbb{C}$:

• $1\cdot A=A$;

• ;

• $\lambda \cdot \left( A+B \right)=\lambda \cdot A+\lambda \cdot B$;

• ;

  3. The transpose of a matrix A= with m rows and n columns is a matrix written ${}^{t}A={{\left( {{b}_{ij}} \right)}_{\begin{smallmatrix} i=1,n \\ j=1,m \end{smallmatrix}}}$ with n rows and m columns, where b_ij = a_ij, $(\forall ) i=\overline{1,n}, j=\overline{1,m}\text{.}$

  4. Matrix multiplication

Let and $B\in {{M}_{n,p}}(\mathbb{C})$, then , where $\displaystyle {{c}_{ij}}=\sum\limits_{k=1}^{n}{{{a}_{ik}}\cdot {{b}_{kj}}}$, is their product.

Properties:

• $\left( A\cdot B \right)\cdot C=A\cdot \left( B\cdot C \right)$ (associativity);

• $A\cdot {{I}_{n}}={{I}_{n}}\cdot A$ (identity element). The identity matrix ${{I}_{n}}=\left( \begin{aligned} & 1\text{ }0 \cdots 0 \\ & 0\text{ }1 \cdots 0 \\ & \cdots \cdots \cdots \\ & 0\text{ }0 \cdots 1 \\ \end{aligned} \right)$ in ${{M}_{n}}(\mathbb{C})$;

• $\left( A+B \right)\cdot C=A\cdot C+B\cdot C;$

• $A\cdot \left( B+C \right)=A\cdot B+A\cdot C$.

Determinants

Let ${{M}_{n}}(\mathbb{C})$ be the set of square matrices of order n with entries in $\mathbb{C}$: $A=\left( \begin{aligned} & {{a}_{11}}\text{ }{{a}_{12}} \cdots {{a}_{1n}} \\ & {{a}_{21}}\text{ }{{a}_{22}} \cdots {{a}_{2n}} \\ & \cdots \text{ }\cdots \text{ }\cdots \text{ }\cdots \\ & {{a}_{n1}}\text{ }{{a}_{n2}} \cdots {{a}_{nn}} \\ \end{aligned} \right)$, $A\in {{M}_{n}}(\mathbb{C}).$

The determinant of the order-2 matrix $\left( \begin{matrix} {{a}_{11}} & {{a}_{12}} \\ {{a}_{21}} & {{a}_{22}} \\ \end{matrix} \right)$ is $\Delta =\left| \begin{matrix} {{a}_{11}} & {{a}_{12}} \\ {{a}_{21}} & {{a}_{22}} \\ \end{matrix} \right|$=a₁₁a₂₂ – a₁₂a₂₁.

The determinant of the order-3 matrix A=$\left( \begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\ \end{matrix} \right)$ is the number

$\Delta =\det A=\left| \begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\ \end{matrix} \right|$=${{a}_{11}}{{a}_{22}}{{a}_{33}}+{{a}_{21}}{{a}_{12}}{{a}_{13}}+{{a}_{31}}{{a}_{12}}{{a}_{23}}-{{a}_{31}}{{a}_{22}}{{a}_{13}}-{{a}_{32}}{{a}_{23}}{{a}_{11}}-{{a}_{33}}{{a}_{21}}{{a}_{12}}$.

The rule of minors, or expanding the determinant along a row or column.

Choose a row or a column and multiply each entry a_ij of that row or column by the lower-order determinant obtained by deleting row i and column j, and by (-1)^i+j; adding the resulting products gives the value of the determinant. $\begin{aligned} & \left| \begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\ \end{matrix} \right|={{(-1)}^{1+1}}{{a}_{11}}\left| \begin{matrix} {{a}_{22}} & {{a}_{23}} \\ {{a}_{32}} & {{a}_{33}} \\ \end{matrix} \right|+{{(-1)}^{1+2}}{{a}_{12}}\left| \begin{matrix} {{a}_{21}} & {{a}_{23}} \\ {{a}_{31}} & {{a}_{33}} \\ \end{matrix} \right|+{{(-1)}^{1+3}}{{a}_{13}}\left| \begin{matrix} {{a}_{21}} & {{a}_{22}} \\ {{a}_{31}} & {{a}_{32}} \\ \end{matrix} \right|= \\ & =-{{a}_{21}}\left| \begin{matrix} {{a}_{12}} & {{a}_{13}} \\ {{a}_{32}} & {{a}_{33}} \\ \end{matrix} \right|+{{a}_{22}}\left| \begin{matrix} {{a}_{11}} & {{a}_{13}} \\ {{a}_{31}} & {{a}_{33}} \\ \end{matrix} \right|-{{a}_{23}}\left| \begin{matrix} {{a}_{11}} & {{a}_{12}} \\ {{a}_{31}} & {{a}_{32}} \\ \end{matrix} \right|=... \\ \end{aligned}$

Properties of determinants:

1. The determinant of a matrix equals the determinant of its transpose.

$\det \left( A \right)=\det \left( {}^{t}A \right)$

2. A matrix with a row (or column) whose entries are all 0 has determinant 0.

3. If all entries of a row (or column) of a matrix are multiplied by a number, the value of the determinant is multiplied by that number.

4. If to all entries of a row (respectively column) we add the corresponding entries of another row (respectively column) multiplied by a number, the value of the determinant does not change.

5. If a matrix has two proportional rows (respectively columns), then the determinant is zero.

6. If two rows (or two columns) of a square matrix are interchanged, the value of the determinant is multiplied by -1.

7. The determinant of the product of two matrices of the same order equals the product of the determinants of those matrices.

$\det \left( A\cdot B \right)=\det \left( A \right)\cdot \det \left( B \right)$

8. If a row (respectively column) of the determinant of a matrix is a linear combination of the other rows (respectively columns), that determinant is zero.

The value $\Delta$ of the determinant of the matrix associated with a system determines whether the system is consistent: if $\Delta$≠0, the system is consistent with a unique solution; if $\Delta$=0, then the system may be inconsistent or consistent and indeterminate.

Linear systems

We write ${{a}_{ij}}$ for the coefficients, ${{x}_{i}}$ for the unknowns, ${{b}_{i}}$ for the constant terms, $m$ for the number of equations, $n$ for the number of unknowns, r for the rank of the matrix A or the rank of the system, and $\overline{\text{A}}$ for the augmented matrix.

A system is consistent with a unique solution if:

1. r=m=n and $\det A=\Delta \ne 0$;

2. r=n<m $rang\overline{\text{A}}=r$.

A system is consistent and indeterminate if:

1. r=m<n;

2. r<min(m,n) and $rang\overline{\text{A}}=rang\text{A}=r.$

A system is inconsistent if $\text{r}\le \text{min}(m,n)$ and $rang\overline{\text{A}}=r+1.$

Cramer's method.

Let S be a linear system with unknowns x_i and $\Delta$ its determinant. Suppose $\Delta$≠0 and r=m=n. Write ${{\Delta }_{{{x}_{i}}}}$ for the determinant obtained from $\Delta$ by replacing the column of coefficients of the unknown x_i with the column of constant terms.

The solution of the linear system $\left\{ \begin{aligned} & {{a}_{11}}{{x}_{1}}+{{a}_{12}}{{x}_{2}}={{b}_{1}} \\ & {{a}_{21}}{{x}_{1}}+{{a}_{22}}{{x}_{2}}={{b}_{2}} \\ \end{aligned} \right.$ is x₁ =$\frac{{{\Delta }_{x1}}}{\Delta }$ and x₂=$\frac{{{\Delta }_{{{x}_{2}}}}}{\Delta }$.

$A=\left( \begin{matrix} {{a}_{11}} & {{a}_{12}} \\ {{a}_{21}} & {{a}_{22}} \\ \end{matrix} \right)$, $\overline{\text{A}}=\left( \begin{matrix} {{a}_{11}} & {{a}_{12}} & {{b}_{1}} \\ {{a}_{21}} & {{a}_{22}} & {{b}_{2}} \\ \end{matrix} \right)$

The solution of the linear system $\left\{ \begin{aligned} & {{a}_{11}}{{x}_{1}}+{{a}_{12}}{{x}_{2}}+{{a}_{13}}{{x}_{3}}={{b}_{1}} \\ & {{a}_{12}}{{x}_{1}}+{{a}_{22}}{{x}_{2}}+{{a}_{23}}{{x}_{3}}={{b}_{2}} \\ & {{a}_{13}}{{x}_{1}}+{{a}_{23}}{{x}_{2}}+{{a}_{33}}{{x}_{3}}={{b}_{3}} \\ \end{aligned} \right.$ is x₁₌$\frac{{{\Delta }_{x1}}}{\Delta }$, x₂=$\frac{{{\Delta }_{{{x}_{2}}}}}{\Delta }$ and x₃₌$\frac{{{\Delta }_{{{x}_{3}}}}}{\Delta }$.

The solution of the system $\begin{aligned} & \left\{ \begin{aligned} & {{a}_{11}}{{x}_{1}}+{{a}_{12}}{{x}_{2}}+{{a}_{13}}{{x}_{3}}+{{a}_{14}}{{x}_{4}}={{b}_{1}} \\ & {{a}_{12}}{{x}_{1}}+{{a}_{22}}{{x}_{2}}+{{a}_{23}}{{x}_{3}}+{{a}_{24}}{{x}_{4}}={{b}_{2}} \\ & {{a}_{13}}{{x}_{1}}+{{a}_{23}}{{x}_{2}}+{{a}_{33}}{{x}_{3}}+{{a}_{34}}{{x}_{4}}={{b}_{3}} \\ & {{a}_{14}}{{x}_{1}}+{{a}_{24}}{{x}_{2}}+{{a}_{34}}{{x}_{3}}+{{a}_{44}}{{x}_{4}}={{b}_{4}} \\ \end{aligned} \right. \\ & \\ \end{aligned}$ is x₁₌$\frac{{{\Delta }_{{{x}_{1}}}}}{\Delta }$, x₂=$\frac{{{\Delta }_{{{x}_{2}}}}}{\Delta }$, x₃₌$\frac{{{\Delta }_{{{x}_{3}}}}}{\Delta }$ and x₄₌$\frac{{{\Delta }_{{{x}_{4}}}}}{\Delta }$.

A homogeneous linear system has all determinants ${{\Delta }_{{{x}_{i}}}}$ zero; it therefore always admits at least the zero solution (x₁,x₂,x₃)=(0,0,0). Such a system also admits non-zero solutions if $\Delta$=0.

Systems of homogeneous equations

Such a system has the form:

$\left\{ \begin{aligned} & {{a}_{1}}{{x}^{2}}+{{b}_{1}}xy+{{c}_{1}}{{y}^{2}}={{d}_{1}} \\ & {{a}_{2}}{{x}^{2}}+{{b}_{2}}xy+{{c}_{2}}{{y}^{2}}={{d}_{2}} \\ \end{aligned} \right.$

Suppose ${{d}_{1}}\ne 0\text{ i }{{d}_{2}}\ne 0$; in this case we multiply the first equation and add it to the second so as to obtain an equivalent system of the form:

$\left\{ \begin{aligned} & {{a}_{1}}{{x}^{2}}+{{b}_{1}}xy+{{c}_{1}}{{y}^{2}}={{d}_{1}} \\ & {{a}_{3}}{{x}^{2}}+{{b}_{3}}xy+{{c}_{3}}{{y}^{2}}=0 \\ \end{aligned} \right.$

Since ${{d}_{1}}\ne 0$ the system does not have the solution $x=0$ and $y=0$. We divide the second equation by ${{x}^{2}}$ and obtain the second-degree equation in $\frac{y}{x}$: ${{a}_{3}}+{{b}_{3}}\frac{y}{x}+{{c}_{3}}{{\left( \frac{y}{x} \right)}^{2}}=0$ which, by the substitution $\frac{y}{x}=t$, gives: ${{c}_{3}}{{t}^{2}}+{{b}_{3}}t+{{a}_{3}}=0$. Solving this equation generally gives two values ${{t}_{1}}\text{ i }{{t}_{2}}$, that is $\frac{y}{x}={{t}_{1}}$ and $\frac{y}{x}={{t}_{2}}$.

Solving the original system is equivalent to solving the following two systems:

$\left\{ \begin{aligned} & y={{t}_{1}}x \\ & {{a}_{1}}{{x}^{2}}+{{b}_{1}}xy+{{c}_{1}}{{y}^{2}}={{d}_{1}} \\ \end{aligned} \right.$ and $\left\{ \begin{aligned} & y={{t}_{2}}x \\ & {{a}_{1}}{{x}^{2}}+{{b}_{1}}xy+{{c}_{1}}{{y}^{2}}={{d}_{1}} \\ \end{aligned} \right.$

Systems of symmetric equations

An equation in two unknowns is called symmetric if replacing $x$ by $y$ and $y$ by $x$ leaves the equation unchanged.

$\left\{ \begin{aligned} & xy+x+y=39 \\ & {{x}^{2}}-xy+{{y}^{2}}=63 \\ \end{aligned} \right.$

Systems of symmetric equations are solved as follows: introduce the auxiliary unknowns $s$ and $p$ given by the relations $s=x+y$ and $p=x\cdot y$.

By introducing these new unknowns $s$ and $p$, in very many cases the original system reduces to a system consisting of one first-degree equation and one second-degree equation in the unknowns $s$ and $p$.

Once the unknowns $s$ and $p$ have been found, one forms the equation ${{t}^{2}}-s\cdot t+p=0$

Since the system considered is symmetric in $x$ and $y$, the solution set of the system is:

$S=\{({{t}_{1}},{{t}_{2}});({{t}_{2}},{{t}_{1}})\}$ if ${{t}_{1}}\ne {{t}_{2}}$, or $S=\{({{t}_{1}},{{t}_{2}})\}$ if ${{t}_{1}}={{t}_{2}}$.

The inverse of a matrix

Let A$\in {{\Mu }_{n}}(\mathbb{C})$ be a square matrix of order n with coefficients in $\mathbb{C}$. The matrix A is invertible if and only if detA≠0. The inverse of the matrix A is A^-1=$\frac{1}{\det A}{{A}^{*}}$, where A^* is obtained by replacing each entry of the transposed matrix ${}^{t}A$ with its cofactor ${{\delta }_{ij}}={{(-1)}^{i+j}}{{d}_{ij}}$, 1*≤i≤n,* 1*≤j≤m*, where ${{d}_{ij}}$ is the minor of the entry ${{a}_{ij}}$ of ${}^{t}A$ (the determinant obtained from ${}^{t}A$ by deleting row i and column j).

$A\cdot {{A}^{-1}}={{I}_{n}},$ ${{I}_{n}}\in {{\Mu }_{n}}(\mathbb{C}),$ ${{I}_{n}}$ - the identity matrix.

Matrix equations

A linear system can be expressed in matrix form as: AX=B, where A is the matrix of coefficients of the unknowns, X is the matrix of unknowns (a column matrix) and B is the matrix of constant terms (a column matrix). If the matrix is invertible we have

X=A^-1B.

If the matrix A$\in {{\Mu }_{m,n}}(\mathbb{C})$ is not zero, there is a natural number r≤min{m,n} such that at least one minor of order r (formed at the intersection of r rows and r columns of the matrix) is non-zero, while all determinants of order greater than r (if any) are zero. This number r is called the rank of the matrix.

The rank of a matrix is unchanged if:

• A multiple of one row (column) is added to another row (column);

• Rows (columns) are interchanged.

The Kronecker–Capelli theorem. A system of linear equations is consistent if and only if the rank of the system matrix equals the rank of the augmented matrix, formed from the system matrix with the column of constant terms added.

Rouché's theorem. A system of linear equations is consistent if and only if all the characteristic determinants are zero. The characteristic determinants are obtained by adding a row and a column from the augmented matrix to a minor with non-zero determinant.

The matrix of rotations in the plane.

Let $\theta \in [0,2\pi )$. The rotation with centre O through an angle of measure $\theta$ is the geometric transformation ${{R}_{\theta }}:P\to P, {{R}_{\theta }}(0)=0$, and ${{R}_{\theta }}(M)={M}'$, $(\forall )M\in P$ with the property that $OM=O{M}'$ and the measure of the angle $\overset\frown{MO{M}'}$ is $\theta$.

The analytic description of the rotation is given by the relations:

$\left\{ \begin{aligned} & {x}'=x\cos (\theta )-y\sin (\theta ) \\ & {y}'=x\sin (\theta )+y\cos (\theta ) \\ \end{aligned} \right.$.

With these, the rotation ${{R}_{\theta }}$ is the function ${{R}_{\theta }}:{{\mathbb{R}}^{2}}\to {{\mathbb{R}}^{2}}$, ${{R}_{\theta }}((x,y))=({x}',{y}')$ where ${x}', {y}'$ are given by the relations above. The matrix associated with the linear map ${{R}_{\theta }}$ is:

$A=\left( \begin{matrix} \cos (\theta ) & -\sin (\theta ) \\ \sin (\theta ) & \cos (\theta ) \\ \end{matrix} \right)$