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Stefan Kohl ♦
Stefan Kohl ♦
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A real symmetric positive semi-definite matrix $A$ can be decomposed in the form

$A = P^TLDL^TP$,

where $P$ is a permutation matrix, $L$ is a lower unit triangular matrix and $D$ is a diagonal matrix.

Questions

(a) Is this decomposition unique?

(b) Let $B = \Pi^TA\Pi$, where $\Pi$ is a permutation matrix. It follows that

$B = \Pi^TP^TLDL^TP\Pi = Q^TLDL^TQ$,

where $Q = \Pi P$ is a permutation matrix, too. If the LDLT decomposition is unique, this means that it is invariant to simultaneous permutations of rows and columns of a matrix, right?

A real symmetric positive semi-definite matrix $A$ can be decomposed in the form

$A = P^TLDL^TP$,

where $P$ is a permutation matrix, $L$ is a lower unit triangular matrix and $D$ is a diagonal matrix.

Questions

(a) Is this decomposition unique?

(b) Let $B = \Pi^TA\Pi$, where $\Pi$ is a permutation matrix. It follows that

$B = \Pi^TP^TLDL^TP\Pi = Q^TLDL^TQ$,

where $Q = \Pi P$ is a permutation matrix too. If the LDLT decomposition is unique, this means that it is invariant to simultaneous permutations of rows and columns of a matrix, right?

A real symmetric positive semi-definite matrix $A$ can be decomposed in the form

$A = P^TLDL^TP$,

where $P$ is a permutation matrix, $L$ is a lower unit triangular matrix and $D$ is a diagonal matrix.

Questions

(a) Is this decomposition unique?

(b) Let $B = \Pi^TA\Pi$, where $\Pi$ is a permutation matrix. It follows that

$B = \Pi^TP^TLDL^TP\Pi = Q^TLDL^TQ$,

where $Q = \Pi P$ is a permutation matrix, too. If the LDLT decomposition is unique, this means that it is invariant to simultaneous permutations of rows and columns of a matrix, right?

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user3749105
user3749105
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Uniqueness and invariance of the LDLT decomposition

A real symmetric positive semi-definite matrix A$A$ can be decomposed in the form

$A = P^TLDL^TP$,

where $P$ is a permutation matrix, $L$ is a lower unit triangular matrix and $D$ is a diagonal matrix.

Questions

(a) Is this decomposition unique?

(b) Let $B = \Pi^TA\Pi$, where $\Pi$ is a permutation matrix. It follows that

$B = \Pi^TP^TLDL^TP\Pi = Q^TLDL^TQ$,

where $Q = \Pi P$ is a permutation matrix too. If the LDLT decomposition is unique, this means that it is invariant to simultaneous permutations of rows and columns of a matrix, right?

Uniqueness and invariance of LDLT decomposition

A real symmetric positive semi-definite matrix A can be decomposed in the form

$A = P^TLDL^TP$,

where $P$ is a permutation matrix, $L$ is a lower unit triangular matrix and $D$ is a diagonal matrix.

Questions

(a) Is this decomposition unique?

(b) Let $B = \Pi^TA\Pi$, where $\Pi$ is a permutation matrix. It follows that

$B = \Pi^TP^TLDL^TP\Pi = Q^TLDL^TQ$,

where $Q = \Pi P$ is a permutation matrix too. If the LDLT decomposition is unique, this means that it is invariant to simultaneous permutations of rows and columns of a matrix, right?

Uniqueness and invariance of the LDLT decomposition

A real symmetric positive semi-definite matrix $A$ can be decomposed in the form

$A = P^TLDL^TP$,

where $P$ is a permutation matrix, $L$ is a lower unit triangular matrix and $D$ is a diagonal matrix.

Questions

(a) Is this decomposition unique?

(b) Let $B = \Pi^TA\Pi$, where $\Pi$ is a permutation matrix. It follows that

$B = \Pi^TP^TLDL^TP\Pi = Q^TLDL^TQ$,

where $Q = \Pi P$ is a permutation matrix too. If the LDLT decomposition is unique, this means that it is invariant to simultaneous permutations of rows and columns of a matrix, right?

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user3749105
user3749105
  • 101
  • 5

Uniqueness and invariance of LDLT decomposition

A real symmetric positive semi-definite matrix A can be decomposed in the form

$A = P^TLDL^TP$,

where $P$ is a permutation matrix, $L$ is a lower unit triangular matrix and $D$ is a diagonal matrix.

Questions

(a) Is this decomposition unique?

(b) Let $B = \Pi^TA\Pi$, where $\Pi$ is a permutation matrix. It follows that

$B = \Pi^TP^TLDL^TP\Pi = Q^TLDL^TQ$,

where $Q = \Pi P$ is a permutation matrix too. If the LDLT decomposition is unique, this means that it is invariant to simultaneous permutations of rows and columns of a matrix, right?