Matrix calculus

Learning tasks

A) The space \(M_{n\virgule p}(\mathbb{K})\)
1) Computing a linear combination	Ex 1 Ex 2 Ex 3
2) Decomposing on the elementary matrices	Ex 4 Ex 5 Ex 6 Ex 7
B) Matrix product
3) Computing a matrix product	Ex 8 Ex 9 Ex 10 Ex 11
4) Identifying non-commutativity and zero divisors	Ex 12 Ex 13 Ex 14 Ex 15
5) Computing matrix powers	Ex 16 Ex 17 Ex 18
C) Transpose\(\virgule\) symmetric and antisymmetric matrices
6) Manipulating the transpose	Ex 19 Ex 20 Ex 21
7) Working with symmetric and antisymmetric matrices	Ex 22 Ex 23 Ex 24
D) Diagonal and triangular matrices
8) Computing products of diagonal and triangular matrices	Ex 25 Ex 26 Ex 27 Ex 28
E) Trace
9) Using the trace	Ex 29 Ex 30 Ex 31
F) Invertible matrices
10) Inverting a \(2 \times 2\) matrix	Ex 32 Ex 33 Ex 34
11) Using \((A B)^{-1}\) and \((A^\top)^{-1}\)	Ex 35 Ex 36 Ex 37
G) Elementary operations and computing the inverse
12) Writing the matrix of an elementary row operation	Ex 38 Ex 39 Ex 40
13) Inverting a matrix by row operations	Ex 41 Ex 42 Ex 43 Ex 44 Ex 45
14) Inverting a triangular matrix	Ex 46 Ex 47 Ex 48

Lesson
Text book

Exercises Correction

Conventions

Throughout this exercise sheet, unless otherwise stated, \(\mathbb{K}\) denotes either \(\mathbb{R}\) or \(\mathbb{C}\) and \(n, p, q\) are positive integers. The notation \(M_{n, p}(\mathbb{K})\) is the set of \(n \times p\) matrices with coefficients in \(\mathbb{K}\); \(M_n(\mathbb{K})\) is the square case. \(I_n\) is the identity matrix of size \(n\), \(E_{ij}\) the elementary matrix with a single \(1\) at position \((i, j)\), \(\mathrm{tr}\) the trace, \(A^\top\) the transpose, \(\mathrm{GL}_n(\mathbb{K})\) the group of invertible matrices. The three elementary row operations are written \(L_i \leftrightarrow L_j\) (swap), \(L_i \leftarrow \lambda L_i\) (scaling, \(\lambda \ne 0\)), \(L_i \leftarrow L_i + \lambda L_j\) (transvection, \(i \ne j\)); same notation for columns with \(C\) instead of \(L\).

A) The space \(M_{n\virgule p}(\mathbb{K})\)
    1) Computing a linear combination Ex 1 Ex 2 Ex 3
    2) Decomposing on the elementary matrices Ex 4 Ex 5 Ex 6 Ex 7
B) Matrix product
    3) Computing a matrix product Ex 8 Ex 9 Ex 10 Ex 11
    4) Identifying non-commutativity and zero divisors Ex 12 Ex 13 Ex 14 Ex 15
    5) Computing matrix powers Ex 16 Ex 17 Ex 18
C) Transpose\(\virgule\) symmetric and antisymmetric matrices
    6) Manipulating the transpose Ex 19 Ex 20 Ex 21
    7) Working with symmetric and antisymmetric matrices Ex 22 Ex 23 Ex 24
D) Diagonal and triangular matrices
    8) Computing products of diagonal and triangular matrices Ex 25 Ex 26 Ex 27 Ex 28
E) Trace
    9) Using the trace Ex 29 Ex 30 Ex 31
F) Invertible matrices
    10) Inverting a \(2 \times 2\) matrix Ex 32 Ex 33 Ex 34
    11) Using \((A B)^{-1}\) and \((A^\top)^{-1}\) Ex 35 Ex 36 Ex 37
G) Elementary operations and computing the inverse
    12) Writing the matrix of an elementary row operation Ex 38 Ex 39 Ex 40
    13) Inverting a matrix by row operations Ex 41 Ex 42 Ex 43 Ex 44 Ex 45
    14) Inverting a triangular matrix Ex 46 Ex 47 Ex 48

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