[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.1.3

Use the QR decomposition to prove Hadamard‘s inequality: if $X=(x_1,\cdots,x_n)$, then $$\bex |\det X|\leq \prod_{j=1}^n \sen{x_j}. \eex$$ Equality holds here if and only if the $x_j$ are mutually orthogonal or some $x_j$ are zero. 

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.1.2

Let $X$ be nay basis of $\scrH$ and let $Y$ be the basis biorthogonal to it. Using matrix multiplication, $X$ gives a linear transformation from $\bbC^n$ to $\scrH$. The inverse of this is given by $Y^*$. In the special case when $X$ is orthonormal (so that $Y=X$), this transformation is inner-product preserving if the standard inner product is used on $\bbC^n$. \eex$$

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.1.1

Given any $k$-tupel of linearly independent vectors $X$ as above, there exists a $k$-tuple $Y$ biorthognal to it. If $k=n$, this $Y$ is unique. \eex$$

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