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Matrix algebra

Theorem: let \(A, B\) and \(C\) be matrices and \(\alpha\) and \(\beta\) be scalars. Each of the following statements is valid

  1. \(A + B = B + A\),
  2. \((A + B) + C = A + (B + C)\),
  3. \((AB)C = A(BC)\),
  4. \(A(B + C) = AB + AC\),
  5. \((A + B)C = AC + BC\),
  6. \((\alpha \beta) A = \alpha(\beta A)\),
  7. \(\alpha (AB) = (\alpha A)B = A (\alpha) B\),
  8. \((\alpha + \beta)A = \alpha A + \beta A\),
  9. \(\alpha (A + B) = \alpha A + \alpha B\).
Proof:

Will be added later.

In the case where an \(n \times n\) matrix \(A\) is multiplied by itself \(k\) times it is convenient to use exponential notation: \(AA \cdots A = A^k\).

Definition: the \(n \times n\) identity matrix is the matrix \(I = (\delta_{ij})\), where

\[ \delta_{ij} = \begin{cases} 1 &\text{ if } i = j, \\ 0 &\text{ if } i \neq j.\end{cases} \]

Obtaining for the multiplication of a \(n \times n\) matrix \(A\) with the identitiy matrix; \(A I = A\).

Definition: an \(n \times n\) matrix \(A\) is said to be nonsingular or invertible if there exists a matrix \(A^{-1}\) such that \(AA^{-1} = A^{-1}A = I\). The matrix \(A^{-1}\) is said to be a multiplicative inverse of \(A\).

If \(B\) and \(C\) are both multiplicative inverses of \(A\) then

\[ B = BI = B(AC) = (BA)C = IC = C, \]

thus a matrix can have at most one multiplicative inverse.

Definition: an \(n \times n\) matrix is said to be singular if it does not have a multiplicative inverse.

Or similarly, an \(n \times n\) matrix \(A\) is singular if \(A \mathbf{x} = \mathbf{0}\) for some non trivial \(\mathbf{x} \in \mathbb{R}^n \backslash \{\mathbf{0}\}\). For a nonsingular matrix \(A\), \(\mathbf{x} = \mathbf{0}\) is the only solution to \(A \mathbf{x} = \mathbf{0}\).

Theorem: if \(A\) and \(B\) are nonsingular \(n \times n\) matrices, then \(AB\) is also nonsingular and

\[ (AB)^{-1} = B^{-1} A^{-1}. \]
Proof:

Let \(A\) and \(B\) be nonsingular \(n \times n\) matrices. If we suppose \(AB\) is nonsingular and \((AB)^{-1} = B^{-1} A^{-1}\) we have

\[ (AB)^{-1}AB = (B^{-1} A^{-1})AB = B^{-1} (A^{-1} A) B = B^{-1} B = I, \\ AB(AB)^{-1} = AB(B^{-1} A^{-1}) = A (B B^{-1}) A^{-1} = A A^{-1} = I. \]

Theorem: let \(A\) be a nonsingular \(n \times n\) matrix, the inverse of \(A\) given by \(A^{-1}\) is nonsingular.

Proof:

Let \(A\) be a nonsingular \(n \times n\) matrix, \(A^{-1}\) its inverse and \(\mathbf{x} \in \mathbb{R}^n\) a vector. Suppose \(A^{-1} \mathbf{x} = \mathbf{0}\) then

\[ \mathbf{x} = I \mathbf{x} = (A A^{-1}) \mathbf{x} = A(A^{-1} \mathbf{x}) = \mathbf{0}. \]

Theorem: let \(A\) be a nonsingular \(n \times n\) matrix then the solution of the system \(A\mathbf{x} = \mathbf{b}\) is \(\mathbf{x} = A^{-1} \mathbf{b}\) with \(\mathbf{x}, \mathbf{b} \in \mathbb{R}^n\).

Proof:

Let \(A\) be a nonsingular \(n \times n\) matrix, \(A^{-1}\) its inverse and \(\mathbf{x}, \mathbf{b} \in \mathbb{R}^n\) vectors. Suppose \(\mathbf{x} = A^{-1} \mathbf{b}\) then we have

\[ A \mathbf{x} = A (A^{-1} \mathbf{b}) = (A A^{-1}) \mathbf{b} = \mathbf{b}. \]

Corollary: the system \(A \mathbf{x} = \mathbf{b}\) of \(n\) linear equations in \(n\) unknowns has a unique solution if and only if \(A\) is nonsingular.

Proof:

The proof follows from the above theorem.

Theorem: let \(A\) and \(B\) be matrices and \(\alpha\) and \(\beta\) be scalars. Each of the following statements valid

  1. \((A^T)^T = A\),
  2. \((\alpha A)^T = \alpha A^T\),
  3. \((A + B)^T = A^T + B^T\),
  4. \((AB)^T = B^T A^T\).
Proof:

Will be added later.