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The volume of the unit ball for the spectral norm in nxn real matrices is given by the formula

$$ c_n \int\limits_{[-1,1]^n} \prod_{i < j} |x_i^2-x_j^2| dx_1\dots dx_n $$

where $c_n = n! 4^{-n} \prod_{k=1}^n v_k^2$

and $v_k=\pi^{k/2}/\Gamma(1+k/2)$ is the volume of the unit ball in R^n.

A much more general formula for calculating all kind of similar quantities appears e.g. here (Lemma 1). The proof is by applying the SVD decomposition as a change of variables.

The first values are

• 2/3 π² for 2x2 matrices
• 8/45 π⁴ for 3x3 matrices
• 4/1575 π⁸ for 4x4 matrices ...

There might be a closed formula for the integral above. Edit : such a formula now appears in Armin's answer!!

The volume of the unit ball for the spectral norm in nxn real matrices is given by the formula

$$ c_n \int\limits_{[-1,1]^n} \prod_{i < j} |x_i^2-x_j^2| dx_1\dots dx_n $$

where $c_n = n! 4^{-n} \prod_{k=1}^n v_k^2$

and $v_k=\pi^{k/2}/\Gamma(1+k/2)$ is the volume of the unit ball in R^n.

A much more general formula for calculating all kind of similar quantities appears e.g. here (Lemma 1). The proof is by applying the SVD decomposition as a change of variables.

The first values are

• 2/3 π² for 2x2 matrices
• 8/45 π⁴ for 3x3 matrices
• 4/1575 π⁸ for 4x4 matrices ...

There might be a closed formula for the integral above. Edit : such a formula now appears in Armin's answer!!

The volume of the unit ball for the spectral norm in nxn real matrices is given by the formula

$$ c_n \int\limits_{[-1,1]^n} \prod_{i < j} |x_i^2-x_j^2| dx_1\dots dx_n $$

where $c_n = n! 4^{-n} \prod_{k=1}^n v_k^2$

and $v_k=\pi^{k/2}/\Gamma(1+k/2)$ is the volume of the unit ball in R^n.

A much more general formula for calculating all kind of similar quantities appears e.g. here (Lemma 1). The proof is by applying the SVD decomposition as a change of variables.

The first values are

• 2/3 π² for 2x2 matrices
• 8/45 π⁴ for 3x3 matrices
• 4/1575 π⁸ for 4x4 matrices ...

There might be a closed formula for the integral above. Edit : such a formula appears in Armin's post below !!

The volume of the unit ball for the spectral norm in nxn real matrices is given by the formula

$$ c_n \int\limits_{[-1,1]^n} \prod_{i < j} |x_i^2-x_j^2| dx_1\dots dx_n $$

where $c_n = n! 4^{-n} \prod_{k=1}^n v_k^2$

and $v_k=\pi^{k/2}/\Gamma(1+k/2)$ is the volume of the unit ball in R^n.

A much more general formula for calculating all kind of similar quantities appears e.g. here (Lemma 1). The proof is by applying the SVD decomposition as a change of variables.

The first values are

• 2/3 π² for 2x2 matrices
• 8/45 π⁴ for 3x3 matrices
• 4/1575 π⁸ for 4x4 matrices ...

There might be a closed formula for the integral above. Edit : such a formula now appears in Armin's answer!!

The volume of the unit ball for the spectral norm in nxn real matrices is given by the formula

c_n int_{[-1,1]^n} product_{i < j} |x_i^2-x_j^2| dx_1...dx_n

where c_n = n! 4^{-n} product_{k=1}^n v_k^2

and v_k=pi^{k/2}/Gamma(1+k/2) is the volume of the unit ball in R^n.

A much more general formula for calculating all kind of similar quantities appears e.g. here (Lemma 1). The proof is by applying the SVD decomposition as a change of variables.

The first values are

• 2/3 pi^2 for 2x2 matrices
• 8/45 pi^4 for 3x3 matrices
• 4/1575 pi^8 for 4x4 matrices ...

There might be a closed formula for the integral above. Edit : such a formula appears in Armin's post below !!

The volume of the unit ball for the spectral norm in nxn real matrices is given by the formula

$$ c_n \int\limits_{[-1,1]^n} \prod_{i < j} |x_i^2-x_j^2| dx_1\dots dx_n $$

where $c_n = n! 4^{-n} \prod_{k=1}^n v_k^2$

and $v_k=\pi^{k/2}/\Gamma(1+k/2)$ is the volume of the unit ball in R^n.

A much more general formula for calculating all kind of similar quantities appears e.g. here (Lemma 1). The proof is by applying the SVD decomposition as a change of variables.

The first values are

• 2/3 π² for 2x2 matrices
• 8/45 π⁴ for 3x3 matrices
• 4/1575 π⁸ for 4x4 matrices ...

There might be a closed formula for the integral above. Edit : such a formula appears in Armin's post below !!