मैट्रिक्स विकर्णीकरण और चोलेस्की अपघटन: सममित मैट्रिक्स का त्वरित सारणिक

Direct Answer / AI Overview: Matrix Diagonalization expresses a square matrix $A$ in terms of its eigenvalues and eigenvectors as $A = PDP^{-1}$, where $D$ is a diagonal matrix of eigenvalues and $P$ is the modal matrix of eigenvectors. Under similarity transformation, determinants are invariant: $\det(A) = \det(PDP^{-1}) = \det(P)\det(D)\det(P^{-1}) = \det(D) = \prod_{i=1}^n \lambda_i$. For symmetric positive-definite matrices, Cholesky Decomposition factors $A$ into a lower triangular matrix and its conjugate transpose ($A = LL^T$). Because the determinant of a triangular matrix is the product of its diagonal elements, $\det(A) = \det(L)\det(L^T) = \left(\prod_{i=1}^n l_{ii}\right)^2$, enabling lightning-fast determinant evaluation in machine learning, statistics, and engineering.

Part 1: Matrix Diagonalization and the Eigenvalue Determinant Rule

An $n \times n$ matrix $A$ is diagonalizable if there exists an invertible matrix $P$ and a diagonal matrix $D$ such that:

$$A = P D P^{-1}$$

Where:

• $D = \text{diag}(\lambda_1, \lambda_2, \dots, \lambda_n)$ contains the eigenvalues of $A$.
• $P = [\mathbf{v}_1 \mid \mathbf{v}_2 \mid \dots \mid \mathbf{v}_n]$ contains the corresponding linearly independent eigenvectors.

Determinant Invariance Under Similarity Transformations

Applying the determinant product theorem ($\det(AB) = \det(A)\det(B)$):

$$\det(A) = \det(P \cdot D \cdot P^{-1}) = \det(P) \cdot \det(D) \cdot \det(P^{-1})$$

Since $\det(P^{-1}) = \frac{1}{\det(P)}$:

$$\det(A) = \det(P) \cdot \det(D) \cdot \frac{1}{\det(P)} = \det(D)$$

For any diagonal matrix, the determinant is simply the product of its diagonal elements:

$$\det(A) = \prod_{i=1}^n \lambda_i = \lambda_1 \cdot \lambda_2 \dots \lambda_n$$

[!TIP] The determinant of any square matrix is equal to the product of all its eigenvalues (counted with algebraic multiplicity). If even a single eigenvalue is zero ($\lambda_k = 0$), then $\det(A) = 0$ and the matrix is singular. Read more in our guide on Eigenvalues & Characteristic Polynomials.

Part 2: Cholesky Decomposition for Symmetric Positive-Definite Matrices

In data science, covariance modeling, Kalman filtering, and structural finite-element analysis, matrices are frequently symmetric positive-definite (SPD) ($A = A^T$ and $\mathbf{x}^T A \mathbf{x} > 0$ for all $\mathbf{x} \neq \mathbf{0}$).

For any SPD matrix, the Cholesky Decomposition factors $A$ uniquely as:

$$A = L L^T$$

Where $L$ is a lower triangular matrix with strictly positive real diagonal entries:

$$L = \begin{bmatrix} l_{11} & 0 & 0 \ l_{21} & l_{22} & 0 \ l_{31} & l_{32} & l_{33} \end{bmatrix}$$

Fast Determinant Formula via Cholesky Factorization

Because the determinant of a triangular matrix equals the product of its main diagonal:

$$\det(L) = \prod_{i=1}^n l_{ii} = l_{11} \cdot l_{22} \dots l_{nn}$$

Since $\det(L^T) = \det(L)$:

$$\det(A) = \det(L L^T) = \det(L) \cdot \det(L^T) = [\det(L)]^2 = \left( \prod_{i=1}^n l_{ii} \right)^2$$

In logarithmic form (critical for preventing numerical underflow/overflow in statistical software):

$$\ln \det(A) = 2 \sum_{i=1}^n \ln(l_{ii})$$

Step-by-Step Cholesky Worked Example

Let us evaluate the determinant of the symmetric positive-definite matrix $A$:

$$A = \begin{bmatrix} 4 & 12 & -16 \ 12 & 45 & -44 \ -16 & -44 & 93 \end{bmatrix}$$

Step 1: Compute Entries of Lower Triangular Matrix $L$

Using the Cholesky recurrence formulas:

1. First Column:

   • $l_{11} = \sqrt{a_{11}} = \sqrt{4} = 2$
   • $l_{21} = \frac{a_{21}}{l_{11}} = \frac{12}{2} = 6$
   • $l_{31} = \frac{a_{31}}{l_{11}} = \frac{-16}{2} = -8$

2. Second Column:

   • $l_{22} = \sqrt{a_{22} - l_{21}^2} = \sqrt{45 - 6^2} = \sqrt{45 - 36} = \sqrt{9} = 3$
   • $l_{32} = \frac{a_{32} - l_{31}l_{21}}{l_{22}} = \frac{-44 - (-8)(6)}{3} = \frac{-44 + 48}{3} = \frac{4}{3}$ (Let’s verify with exact fractions)

3. Third Column:

   • $l_{33} = \sqrt{a_{33} - (l_{31}^2 + l_{32}^2)} = \sqrt{93 - (64 + \frac{16}{9})} = \sqrt{29 - \frac{16}{9}} = \sqrt{\frac{245}{9}}$

Thus, the determinant is computed directly from the diagonal products:

$$\det(A) = (l_{11} \cdot l_{22} \cdot l_{33})^2$$

Compare this efficiency to standard Gaussian elimination using our Determinant Calculator with Steps.

Comparison: LU vs. Cholesky vs. Eigendecomposition

For general nonsymmetric matrices, our online Determinant Solver employs partial-pivoted LU decomposition to ensure numerical accuracy.