Characterisation of Real Symmetric Positive Definite Matrix/Necessary Condition
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Theorem
Let $A$ be an $n \times n$ positive definite symmetric matrix over $\RR$.
Then:
- there exists an invertible matrix $C$ such that $A = C^\intercal C$.
Proof
Let $A$ be positive definite.
From Real Symmetric Matrix is Orthogonally Diagonalizable:
- there exists an orthogonal matrix $P$ and diagonal matrix $D$ such that $A = P^\intercal D P$.
Further:
- the diagonal entries of $D$ are the eigenvalues of $A$.
From Real Symmetric Positive Definite Matrix has Positive Eigenvalues:
- the diagonal entries of $D$ are positive.
We can therefore construct a real diagonal matrix $S$ by:
- $\paren S_{i j} = \begin{cases} \sqrt {\paren D_{i i} } & i = j \\ 0 & i \ne j \end{cases}$
From Product of Diagonal Matrices is Diagonal, we have:
- $\paren {S^2}_{i j} = \begin{cases} \paren D_{i i} & i = j \\ 0 & i \ne j \end{cases}$
so:
- $S^2 = D$
We also have:
\(\ds \det S\) | \(=\) | \(\ds \prod_{i \mathop = 1}^n \sqrt {\paren D_{i i} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {\prod_{i \mathop = 1}^n \paren D_{i i} }\) | ||||||||||||
\(\ds \) | \(>\) | \(\ds 0\) | as $\paren D_{i i} > 0$ for each $i$ |
We therefore have:
\(\ds \map \det {P^\intercal S P}\) | \(=\) | \(\ds \map \det {P^\intercal} \det S \det P\) | Determinant of Matrix Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\det P}^2 \det S\) | Determinant of Transpose | |||||||||||
\(\ds \) | \(=\) | \(\ds \det S\) | Determinant of Orthogonal Matrix is Plus or Minus One | |||||||||||
\(\ds \) | \(>\) | \(\ds 0\) |
So from Matrix is Invertible iff Determinant has Multiplicative Inverse:
- $P^\intercal S P$ is invertible.
Let $C = P^\intercal S P$.
Then:
\(\ds C^\intercal C\) | \(=\) | \(\ds \paren {P^\intercal S P}^\intercal P^\intercal S P\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds P^\intercal \paren {P^\intercal S}^\intercal P^\intercal S P\) | Transpose of Matrix Product | |||||||||||
\(\ds \) | \(=\) | \(\ds P^\intercal S^\intercal P P^\intercal S P\) | Transpose of Matrix Product | |||||||||||
\(\ds \) | \(=\) | \(\ds P^\intercal S^\intercal S P\) | as $P$ is orthogonal | |||||||||||
\(\ds \) | \(=\) | \(\ds P^\intercal S^2 P\) | Diagonal Matrix is Symmetric | |||||||||||
\(\ds \) | \(=\) | \(\ds P^\intercal D P\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds A\) |
As $C$ is invertible, the proof is complete.
$\blacksquare$