F16 Ortonormala matriser och spektralsatsen (1MA024: Linjär Algebra II (träning))

Vilka är ortonormala

Vilka av följande matriser är ortonormala

Svara alla som korrekta.

$\tfrac{1}{3}\begin{pmatrix} 1 & -2 & 2 \\ -2 & 1 & 2 \\ 2 & 2 & 1 \end{pmatrix}$ $\tfrac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$ $\tfrac{1}{3}\begin{pmatrix} 1 & -2 & 2 \\ -2 & 1 & 2 \\ 2 & 2 & -1 \end{pmatrix}$ $\begin{pmatrix} 0 & -1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}$ $\tfrac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 & 0 \\ 1 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

Typing functional expressions

Functional expressions are largely written as by hand. Functions must be written with parentheses: i.e. sin(x), not sin x.

Some special functions
  • $\sqrt{x}$ is written sqrt(x)
  • $e^x$ is written exp(x)
  • $|x|$ is written abs(x)

Exponents are written using ^: e.g. $2^{x+y}$ is wrtten as 2^(x+y).

Place parentheses where needed: $\dfrac{x+y}{1+x}$ is written as (x+y)/(1+x)

Multiplication sign may but do not have to be explicitly written out: $x\cdot y$ may be written as xy or x*y.

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